This kind of play style was first introduced in original Borderlands with the release of The Secret Armory of General Knoxx and Loyalty COMs. Each BL1 character got 3 of those. Now in Borderlands 2 we were introduced to a new relic mechanic including the Allegiance relics. Also, this play style is not really defined by any in game mechanic other than your own. For me personally, it all starts with this relic. This all being said, you probably ask yourself, why should i play this?
Well, the game is pretty old now and a lot of people already moved on to other games, but there are still a lot of people playing it. There are people like myself, who like to theorize and experiment and get the most out of it with every piece of gear and/or skill and who simply don't want to let it go :). Anyway, playing allegiance is fun, challenging and simply put... different. It can be played from fresh start or end-game, it is a lot of fun in coop, where each of you play different allegiances and you get to share the loot between the party. Depending on your chosen allegiance, type and class, it can be very, very challenging and hard to master... but fun.
You will get to appreciate and use some gear you would otherwise never use (I am looking at you Landscaper and Commerce, and Judge, and Tinderbox.... :)). Finding the synergy and getting used to specific manufacturer traits is the key.

Like I've already said, it should start with the relic. All bonuses on this relic are additive, other than that @Adabiviak sums it up pretty well:

Allegiance relics can buff three different aspects of that character, but only two may be buffed per relic. Note that those relics that buff damage have this damage factored in at a point in the calculation that makes it a bit less damaging than an elemental damage boost of the same value. The benefit is that this damage will come with a secondary perk that may be worth it to the player, and with an allegiance build, you can have a mix of elemental weapons in your loadout, and the relic damage will buff them all. Here is a list of the available buffs per manufacturer. Remember: each relic may come with any two of these:

DAHL: Magazine Size, Recoil Reduction, Burst Delay (Dahl exclusive)
HYPERION: Damage Increase, Reload Speed, Max Accuracy (Hyperion exclusive)
BANDIT: Damage Increase, Reload Speed, Fire Rate
JAKOBS: Accuracy Recovery, Magazine Size, Recoil Reduction
MALIWAN: Damage Increase, Reload Speed, Fire Rate
TEDIORE: Damage Increase, Magazine Size, Recoil Reduction
TORGUE: Magazine Size, Accuracy Recovery, Fire Rate
VLADOF: Damage Increase, Accuracy Recovery, Recoil Reduction

Based on the observation and discussions with some other people on the GBX forums, there are 2 Allegiance types:

A. ABSOLUTE ALLEGIANCE

Absolute allegiance is the one where the player uses only and ONLY the pieces of the equipment made by the same manufacturer. No exceptions! That means if your manufacturer does not produce a COM / grenade for your allegiance, you should not use any equipment in these slots at all.

B. CASUAL ALLEGIANCE

Casual allegiance is the one where player have his own set of rules. In most of the cases, he chooses to use all the guns from the same manufacturer (and/or shields / grenades) while using the COM from the other manufacturer (and/or non-allegiance relic). Also in addition, casual allegiance players are using Anshin and Pangolin as replacements.

The only real drawback with the Allegiance in this game are class mods imo. There are a lot of missing ones for each character, and some of them does not make any sense at all. For example, Psycho have all Bandit COMs, with the exception of Slayer Of Terramorphous, which is Maliwan. Mechromancers Jill of all Trades COM is manufactured by Tediore which gives bonuses to assault rifles, while Tediore does not make any. Also there is some COMs which do not synergize with given characters at all. Each manufacturer is ranked 1-10, 1 being a cakewalk and 10 being super hard. Anyway lets move on to the manufacturers:

Ok, this is not in-depth guide on gear progression, but just a little pointer on when you can grab some piece of the gear you could use for your allegiance. Note that these are based on my observation but it should be like 90% accurate. Starting from LV1:

LV1: Dahl Basic Repeater in Claptrap's Place - Obviously, you will use this one if you plan on doing Dahl allegiance. This gun is unique in game files,
~LV2-3: All pistol brands, including Torgue explosive. Maliwan comes only as incendiary,
~LV2-3: Tediore shields. These are introductory and the first ones you can grab,
~LV3: Some assault rifles (Vladof, Dahl, Jakobs maybe), including grenadiers and rocketeers,
~LV3: Some shotguns (Jakobs, Hyperion),
LV5+: Some non elemental sniper rifles,
LV5+: Torgue shotguns, Tediore,
LV5+: Torgue assault rifles,
~LV4-5: Tediore, Bandit grenade mods (explosive only),
LV7+: SMGs,
LV8+: Class Mods,
LV8+: Maliwan spike shields (incendiary, corrosive, shock),
LV8+: Pangolin turtle shields,
~LV10-11: explosive rocket launchers,
~LV10-11: Vladof AOE grenades (incendiary, corrosive, shock),
~LV10-11: Torgue MIRV grenades,
LV12+: Bandit roid shields,
LV12+: Maliwan nova shields (incendiary, corrosive, shock),
LV12+: shock elemental weapons,
~LV13: Dahl booster shields,
~LV14: Vladof absorb shields,
~LV14: Bouncing betty grenade mods,
LV14+: corrosive, slag elemental weapons,
~LV15-16: Hyperion amplify shields,
LV15+: Relics (incl. allegiance),
LV16+: Anshin adaptive shields,
~LV17: Maliwan transfusion grenades,
LV17+: Hyperion singularity grenades,
LV30+: Jakobs Rough Rider shield.

That's about it. Depending on your allegiance type it could be rough to play without the shield, especially going for Rough Rider. For some reason that DLC scales from LV30, where all other DLCs scale from LV15. This could be a bug imo. Also take note that when you visit certain area throughout the story, the vendors there will always have the same gear types for that level. Only the vendors in Sanctuary scales with your level, the gear as well.
I've made a thread on the official GBX forums where I will attempt to grab Rough Rider at LV30+ while beating the whole game on normal without using any shields as Absolute Jakobs allegiance assassin. I will post only the latest episode here, but you can find the whole journey on the forums here[forums.gearboxsoftware.com].

https://www.youtube.com/watch?v=xnvhae6QLqM

The only real drawback with the Allegiance in this game are class mods imo. There are a lot of missing ones for each character, and some of them does not make any sense at all. For example, Psycho have all Bandit COMs, with the exception of Slayer Of Terramorphous, which is Maliwan. Mechromancers Jill of all Trades COM is manufactured by Tediore which gives bonuses to assault rifles, while Tediore does not make any. Also there are some COMs which do not synergize with given characters at all. Each manufacturer is ranked 1-10, 10 being super-hard and 1 being a cakewalk. Anyway lets move on to the manufacturers:

Almost all Jakobs guns are non-elemental which means that their base damage is higher than those from the same tier elemental guns from other manufacturers. Also, all Jakobs guns have Type B critical hit bonuses which are multiplicative to other (Type A) critical hit bonus. DLCs have introduced some elemental Jakobs guns like Twister, Greed and Stinkpot, explosive Cobra as well as a one and only Jakobs shield: The Rough Rider. Jakobs allegiance is overall one of the hardest and most challenging to play in UVHM simply because of the lack of shield choices and elemental support, including Slag which is almost mandatory in UVHM. Some of the classes have slag support, some of them don't, like Salvador and Krieg for example. In my opinion Zer0 is the best for this allegiance because he has a dedicated sniper rifle tree and Jakobs have some of the best snipers around. Axton can be really awesome with their assault rifles and slag turret, Maya and Gaige are somewhere in the middle because they have some slag options, while Salvador and Krieg are the worst there simply because of theirs lack of slag, although both of them can have some nice damage reduction synergy with the RR and their skills. Using Jakobs while gunzerking is also a downside because of the constant buttons mashing. You need to get very creative when it comes to Jakobs and UVHM.
Also, the difficulty depends on your choice to play casual or absolute allegiance, where casual can be more adjustable, while absolute is even harder with some classes because of no COM and because you can get Rough Rider shield really late in the game.
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PROS: High weapon base damage | Multiplicative Type B critical hit bonus
CONS: Limited elemental support | No grenades | No slag weapons | No rocket launchers
OVERALL DIFFICULTY: 8/10
BEST WITH: ZER0
WORST WITH: SALVADOR
_________________________
CLASS MOD AVAILABILITY:

ZER0: Legendary Hunter | Legendary Sniper | Sniper | Slayer Of Terramorphous
GAIGE: None
AXTON: None
SALVADOR: Renegade
MAYA: None
KRIEG: None

Ah, the Bandit. One of my favorite allegiances. Their guns have terrible reload speeds and almost everything else with the exception of magazine size. The Bandit is also the only manufacturer to produce 5 weapon types, as opposed to 4 like other manufacturers. Also they are the only ones to produce more than one type of grenades, including standard ones, bouncing betties and MIRV grenades, and in different elements. Nevertheless, the best Bandit products are roid shields, which can be put to great use with Zer0, Krieg and DT Gaige. Other characters can make good use of Bandit guns, especially Salvador with gunzerking, high selection of his fire rate skills and with auto-loader making Bandit guns equally efficient. There is not much synergy with Axton and Bandit making him the worst for this allegiance imo, not to mention the fact that he is the only character which don't have a Bandit COM when it comes to absolute allegiance.
_________________________
PROS: Roid shields | Wide weapon and grenade mod selection
CONS: Below average weapon stats
OVERALL DIFFICULTY: 6/10
BEST WITH: KRIEG | ZER0 | GAIGE
WORST WITH: AXTON
_________________________
CLASS MOD AVAILABILITY:

ZER0: Survivor
GAIGE: Punk
AXTON: None
SALVADOR: Devastator | Raider | Legendary Berserker | Slayer Of Terramorphous
MAYA: Banshee | Cleric
KRIEG: All, except Slayer Of Terramorphous

This is one of the harder allegiances, simply because them Booster shields. Of all the shields in the game, booster shields are the weakest imo. Only one legendary shield and he tends to kill you more than your enemies. However if you can master it, you should be fine. The best way to play this allegiance is to be defensive as much as possible. Take cover, aim, shoot, repeat. Their grenades are surprisingly good though. Use them to slag everything before getting out from the cover. I've only played this allegiance with Axton and Salvador so far, where Axton, when playing defensive can be really good with their assault rifles and SMGs. Dahl allegiance relic haves unique trait which reduces burst delay. Once you get one, it will make your Dahl allegiance much much easier and more enjoyable. Salvador as expected, was my worst experience in this game ever, due to no burst fire while zerking. Solid COM support, with the exception being Axton who haves 7 of them.
_________________________
PROS: Burst Fire for those who like it | Almost no Recoil
CONS: No rocket launchers | Booster shields
OVERALL DIFFICULTY: 6/10
BEST WITH: AXTON
WORST WITH: SALVADOR
_________________________
CLASS MOD AVAILABILITY:

ZER0: Infiltrator | Professional
GAIGE: Prodigy
AXTON: Legendary Engineer | Engineer | Ranger | Rifleman | Veteran | Legendary Soldier | Slayer Of Terramorphous
SALVADOR: War Dog
MAYA: None
KRIEG: None

Choosing the right element for the right opponents can make the Maliwans extremely effective despite their generally low bullet damage, especially on UVHM. DOT damage in general is kinda meh although there are exceptions. Large selections of COMs for either casual or absolute allegiance is available. Maya, LBT Gaige and Hellborn Krieg benefits the most from this brand, because of their skills which can boost elemental damage. Some Maliwan guns gets boosted by grenade damage so its somewhat useful on Axton as well. Zer0 does not have any skills which boosts elemental damage in any way, making him my worst choice for this allegiance (also that Maliwan Ninja COM -_-). The only reason why I didn't choose Salvador is because while gunzerking he can slag and shoot at the same time without the need to switch guns.
_________________________
PROS: Lots of slag! | Best elemental support
CONS:
OVERALL DIFFICULTY: 3/10
BEST WITH: KRIEG | GAIGE | MAYA
WORST WITH: ZER0
_________________________
CLASS MOD AVAILABILITY:

ZER0: Legendary Ninja | Ninja | Rogue | Spy
GAIGE: Legendary Catalyst | Catalyst | Zapper | Slayer Of Terramorphous
AXTON: Shock Trooper
SALVADOR: None
MAYA: Legendary Binder | Binder | Legendary Siren | Fox | Witch | Slayer Of Terramorphous
KRIEG: Slayer Of Terramorphous

If you are a casual Borderlands 2 player and you want to try this kind of gameplay, this might be the right allegiance for you. Imo it's the easiest of them all simply because of the Amp shields in general and the famous Bee shield. This allegiance makes UVHM a joke. It's very viable on OP levels as well. Reverse recoil is somewhat tricky cos you will waste a few bullets, but after that hyperions are pinpoint accurate as you hold the trigger down. Again a solid COM support for both allegiance types. The only thing that is somewhat bad, is that Hyperion does not make any rocket launchers, in case that you need one for pick me ups. I chose Zer0 and Maya as best because of the representative COMs. Every other class can do this allegiance just fine as well. Even Krieg!
_________________________
PROS: Pinpoint accuracy | Amp shields
CONS: No rocket launchers
OVERALL DIFFICULTY: 1/10
BEST WITH: ZER0 | MAYA
WORST WITH: KRIEG
_________________________
CLASS MOD AVAILABILITY:

ZER0: Legendary Killer | Killer | Shot
GAIGE: Legendary Roboteer | Roboteer | Legendary Mechromancer
AXTON: Specialist
SALVADOR: None
MAYA: Legendary Cat | Cat
KRIEG: None

Vladof is one of those manufacturers that can make your allegiance gameplay experience a bit more enjoyable. Decent weapon stats with the exceptions of increased fire rate making each and every gun a DPS monster. They eat ammo, but that can be easily solved with the use of their own Absorb shields. No matter which class you pick they all fare very well with this brand. Their AOE grenades are probably, if not the most powerful grenades in the game. I'm sure you all know what sham(drop your fav RL here)ing is. Well, there you go. I can't tell which class is the best with this brand, but I have to pick Salvador here, because he can wield not one, but two of those DPS monsters. I also can't tell, which class is the worst? Either absolute or casual allegiance, all classes have their representative COMs, with the exception of Krieg.
_________________________
PROS: Fire rate! | Absorb shields
CONS: Ammo consumption except for rocket launchers | No SMGs
OVERALL DIFFICULTY: 3/10
BEST WITH: SALVADOR
WORST WITH: KRIEG?
_________________________
CLASS MOD AVAILABILITY:

ZER0: Stalker
GAIGE: Legendary Anarchist | Anarchist
AXTON: Gunner
SALVADOR: Legendary Hoarder | Hoarder | Monk
MAYA: Trickster
KRIEG: None

EXPLOSIONS!? They are slow, yes, but they are DEVASTATING. Because of the slow projectiles speed one need a bit to get used to it, but once you do, everything will just explode in front of you. Almost all guns are boosted by grenade damage bonuses making Torgue even more deadlier with Axton and Krieg. Other classes have their ways around as well. Regarding COMs, Torgue, like Jakobs have poor selection of COMs, making absolute allegiance somewhat more difficult.
_________________________
PROS: EXPLOSIONS! | Splash damage
CONS: No slag | Slow projectiles
OVERALL DIFFICULTY: 3/10
BEST WITH: AXTON | KRIEG
WORST WITH: GAIGE
_________________________
CLASS MOD AVAILABILITY:

ZER0: None
GAIGE: None
AXTON: Grenadier
SALVADOR: Beast | Berserker | Legendary Gunzerker
MAYA: None
KRIEG: None

The only real complaint about this brand I personally have is their lack of COM support. There are only 2 available, and both of them for Gaige, one of them being Jill Of All Trades which absolutely doesn't make any sense imo. The other one is somewhat useful, but there is one major drawback with Gaige and Tediore. There is no synergy at all with their reload mechanic and Anarchy. If you chose not to use Anarchy with this allegiance is somewhat challenging, but still not an easy task to do. Probably best with Axton and/or Krieg because of their grenade damage bonuses. Absolute allegiance have almost no support with this brand.
_________________________
PROS: Reload mechanics is somewhat fun to use
CONS:
OVERALL DIFFICULTY: 5/10
BEST WITH: AXTON
WORST WITH: ZER0
_________________________
CLASS MOD AVAILABILITY:

ZER0: None
GAIGE: Jill Of All Trades | Necromancer
AXTON: None
SALVADOR: None
MAYA: None
KRIEG: None

Coming soon.

• Published the first version of the guide.