The original Crisis on Infinite Earths had this as well — Brainiac and Luthor took advantage of the chaos caused by the Anti-Monitor to gather an army of EVERY known super-villain on the remaining Earths — and unleashed it on the worlds left unprotected when the heroes went off to battle the Big Bad. They also cut down on unnecessary duplicates (Earth-2 Luthor, anyone?).
In Legion of Super-Heroes storyline Legion Of Three Worlds,Superboy-Prime decides to strike at the 31st century that worships Superman by gathering every single Legion foe into a massive Legion of Super-Villains. While they may not all like each other, it's clear the LSV are united in their desire to finally see their long-time enemies dead.
The Sinestro Corps War arc of Green Lantern had something similar. While there was plenty of new baddies created as Sinestro Corps members, the main threat was the team up of the major DCU baddies Sinestro, Superboy-Prime, the Cyborg Superman, Parallex, the Manhunters, and the Anti-Monitor.
Marvel has published two bad-guy-focused mini-series under the banner of Supervillain Team-Up: The first was called M.O.D.O.K.'s 11 and revolves around the titular giant floating head hiring a bunch of D-list supervillains to help him steal a MacGuffin. The second is Dr. Doom and the Masters of Evil, where Doctor Doom manipulates every other supervillain he can find into helping him...well, it takes a while to find out exactly what his goal is. Both involved many, many villain-on-villain betrayals.
These are revivals of the original Super-Villain Team-Up title that Marvel published between 1975 and 1980, which was primarily a showcase for Doctor Doom and Namor the Sub-Mariner.
The Destructix, being a Quirky Miniboss Squad-for hire, have teamed up with anyone willing to pay them, or who earn their loyalty (Mogul, Finitevus, Scourge, etc). The team actually started out as one of these, when the Fearsome Foursome (Mogul's first QMS) teamed up with Snoop Dawg and Drago Wolf, and it solidified into a team.note Though it should be noted that those latter two eventually quit, and have now been replaced by Scourge and Fiona.
In the "World's Collide" crossover with Mega Man, Dr. Eggman teams up with Dr. Wily to conquer both worlds... and surprisingly, they get along pretty well, with the only hiccup being that they have trouble agreeing on what to name their joint creations. At least until the heroes start winning; at that point, the two start plotting to betray and abandon each other.
In the Archie Teenage Mutant Ninja Turtles Adventures comic, a time-displaced Shredder teamed up with future supervillains Armaggon and Verminator X (an anthropomorphic shark and cat, respectively), in order to steal and power a time-machine prototype. In the sixth season of the 4Kids cartoon, Big Bads Sh'Okanabo and Darius Dun form a rare successful alliance when they decide to exchange resources.
Subverted in Turtles Forever, where the 1987 Shredder tries this with his 2003 counterpart, only to get kicked out of the Technodrome and have his resources taken over, leading to an Enemy Mine situation with the 1987, 2003, and Prime Turtles.
Daredevil actually suffered through this more than once. The first time saw Electro organize a group of villains defeated by Daredevil into the "Emissaries of Evil" to try and get revenge on DD, and the second time would occur a few decades later, when Typhoid Mary would gather several latter-day Daredevil villains into the "Daredevil Revenge Squad".
A 1960s Hulk comic had the Mandarin team up with the Sandman. It ended with Mandarin sending the Sandman into a molten vat turning him to glass.
Paste-Pot Pete broke another Human Torch foe, the Wizard, out of jail, however they conflicted due to the Wizard trying to act as leader while Pete wanted them to be equal. They were captured by the Torch, but didn't seem sorry at the team-up ending. The two would nevertheless become founding members of the Frightful Four.
Marvel had a comicbook series called Super-Villain Team-Up, which despite the name was mostly Doctor Doom hanging out with Namor and fighting every two issues.
Several of the Marvel Universe's Big Bads all teamed up in the 1980s Acts of VengeanceCrossover in a large-scale Evil Plan to destroy each others' enemies by setting the heroes up against villains they'd never faced before. In effect, this was a Villain Team Up of almost all the human villains of the Marvel Universe, with a team of Chessmasters manipulating dozens of lesser bad guys for their own personal agenda. The whole scheme falls apart for the simple fact that all of the villains behind the plot are so egomaniacal that they can't stand not being totally in charge, and they end up turning on each other while struggling for power.
Of course, it didn't help that Loki decided that his inner circle should include an UNREPENTANT FORMER NAZI alongside a HOLOCAUST SURVIVOR. Seriously, Loki, we thought you were supposed to be one of the clever Asgardians...
It should be noted that villains can and do form their own teams for reasons beyond simply getting vengeance on a particular hero. The Masters of Evil, one of the Marvel Universe's longest-running supervillain teams, has had various incarnations formed over the years by Big Bads who recruit other villains to share in the profits of the leader's evil scheme. The Serpent Society, long a thorn in the side of Captain America, was a collection of snake-themed villains who essentially formed their own mercenary business, complete with health care coverage, room and board, and a guaranteed "Get out of Jail Free" Card from the team leader, who would use his teleportation powers to free any members who were arrested or captured.
The Batman also did a variation of this in the "Team Penguin" episode, when the Penguin recruits several second-tier Bat villains into a criminal gang as a way of evening the odds against Batman and his sidekicks. The idea is lampshaded at every opportunity, as none of the other villains like the name "Team Penguin" and keep suggesting alternatives, which Penguin shoots down. It's Penguin's vanity that causes the others to abandon the team. At first.
The Fantastic Four had to fight Doctor Doom and the Sub-Mariner at once (in one of the earliest examples), and were saved when Doom betrayed the Sub-Mariner a bit too early, causing him to switch sides. Later, the heroes faced the Frightful Four; a revolving-door group of second-string supervillains that always seemed to betray each other.
Spider-Man had to fight the Sinister Six — six of his enemies united for the sole purpose of killing him. In a variation, they fail not because they don't trust each other, but because they have such big egos that each one has to be the one to deliver the killing blow... so they make him Run the Gauntlet instead of ganging up. A later version did fight Spider-Man as a team, but they've had no better luck. The one time it worked and they had Spidey on the ropes, the Fantastic Four and the Avengers, all at once, swooped in to save the day. Heroes can play that game too.
Another Spider-Man example is the Sinister Syndicate, a group of C-List Spidey villains. The difference between this team and the Sinister Six is that the Syndicate try to focus mainly on making money and would rather avoid having to fight Spider-Man.
There were also the New Enforcers, a semi-motley crew that was so powerful Spider-Man had to whip up some Iron-Man esque armor to be able to face them all at once (the fight ended with the armor being destroyed), and a never named brief team of villains who Spider-Man defeated individually when he was briefly bonded with the Uni-Power who joined up to get revenge on Spidey once he had lost the power. They still lost.
A similar thing happened to Spider-Girl in the Marvel Comics 2 continuity. It was a Run the Gauntlet style, and Spider-Girl was both exhausted and (unknown to herself) depowered before the last fight...so she calls in a favor from pretty much every hero she's ever met. The last villain wisely surrenders.
Which was a continuation Poison Ivy and Harley Quinn's history of teaming up, based on a rare supervillain friendship.
Marvel Comics and DC Comics had several crossovers, where heroes from each group teamed up to fight a crossover Villain Team-Up. Every time, the villains lost by betraying each other.
The Flash's Rogues Gallery is an exception: the villain team, known simply as "The Rogues", works together all the time with practically no problems whatsoever, and is effectively a standing army of super-villains. As the Flash points out to Batman, in Central City the super-villains and organized crime are one and the same.
Not all of Flash's Rogues Gallery falls under "The Rogues" though, Gorilla Grodd won't stand to work with any humans, and the rest of the Rogues REALLY hate Zoom, since he breaks so many of their codes of conduct.
The Hellblazer story "How to Play With Fire" featured several of Constantine's enemies colluding to destroy his life.
In The DCU, the Secret Six are a team of supervillains-for-hire (comprising two Bat-villains, two second-generation versions of JSA villains, and two rotating positions). They'd happily not tangle with any heroes, but don't mind if it happens. They're also usually in the bad books of other villains as well (especially since they won't join the Society). Unusually, they get on reasonably well with each other (mostly), and have a good sense of teamwork.
Recent B.P.R.D. miniseries have focused on a Villain Team-Up between the frog army led by the Black Flame and the slaves of Hyperboria led by the King of Fear. Unlike the average team up, this is proving very successful and has already led to the destruction of one major European city with more carnage on the way.
The main Hellboy series also has a Villain Team-Up between the Fairies, the witches of England, and a growing army of dark creatures. It also has the tacit support of Hell.
In the Mickey Mouse 70th anniversary comic book story "The Past Imperfect," Mickey is captured by a team of his worst comics enemies, including Pegleg Pete, Sylvester Shyster, Eli Squinch, Dr. Vulter, and Profs Ecks, Doublex, and Triplex. Interestingly, the Phantom Blot is not included, as he was being saved for a different story also published at that anniversary.
In the 50th anniversary story that Don Rosa did for Scrooge McDuck, A Little Something Special, his three main foes—The Beagle Boys, Magica DeSpell, and Flintheart Glomgold—all join forces, and are led by Blackheart Beagle. This collaboration is made possible because their interests do not actually overlap: Magica only wants Scrooge's Number One Dime for her spell, the Beagle Boys want the rest of Scrooge's fortune and don't care about a single dime, and Glomgold doesn't want anything from Scrooge per se, just to see his business rival ruined. As a result, the three do not betray one another, but all live up to the deal they made at the start of the team-up. Of course, once the team-up is over, nothing prevents the Beagle Boys and Magica from joining forces to plunder Glomgold...
The European comics have a story arc setting up a new Big Bad, who forced the Beagle Boys, Magica and Phantom Bolt to join him by brainwashing them. This actually led to his undoing as Minnie and Daisy convinced Magica to turn against him and Phantom Bolt was just pretenting to be brainwashed so he could hijack the plot and the two atacked him in the middle of his confrontation with Donald and Mickey.
In the The Incredible Hulk book, there is the Intelligencia, a super villain team made of Mad Scientists. So far they've managed to avoid infighting. In fact, they've been able to work perfectly as a unit.
Not only that, but they apparently worked together for years, and various instances when they fought each other were retconned to be set-up so nobody would suspect a thing.
JSA Classified had an arc focused on the Injustice Society. Unlike some cases, they're very professional about teaming up, with the expressed belief that having a competent pro watching your back beats "every man for himself" any day.
Irredeemable played with this with members of Plutonian's Rogues Gallery offering to join him in his new Face–Heel Turn; he decided to test their loyalty, by offering each of them a button that, when pressed, would render him completely powerless. They betrayed him before he could even finish the sentence. He then revealed they just triggered the destruction of the facility they were in.
This is apparently how various supervillains get to know Max Damage from Incorruptible - in his villainous days they would often team up. One flashback shows Max teaming up with another villain and then them both betraying each other because they just didn't like each other.
The Dark Avengers were in fact a team up of the more villainous members of the Thunderbolts with antiheroes Ares and The Sentry, young and misguided Marvel Boy (who quit the moment he found out what he gotten himself into) and Daken
The Cabal was this between Norman Osborn, Loki, The Hood, Doctor Doom, Namor and Emma Frost and later Taskmaster, with a lot of betrayals and schemes in it.
One of the tie-ins featured a team of supervillains, the Lethal Legion, who decided they don't like Norman being in charge and want to team up against him. It was all a plot set by Norman to make himself look better.
There were also lesser team ups between Doctor Doom and Dracula or Loki, Hela and Mephisto
Hera teamed up with Typhon during that arc and tried to team up with Norman Osborn but, after finding out what her plan was about he decided to send his team against her.
In Supergirl2017 storyline The Girl of No Tomorrow, Emerald Empress recruits four other villains -Selena, Indigo, Magog and Solomon Grundy- to kill Supergirl after destroying her reputation.
Lex: We each have our own nemesis, my friend. You, your Web-slinger — I, my Man of Steel. Still, tomorrow, perhaps...! Doc Ock: "Tomorrow"? Tomorrow, we'll be right here, Luthor — watched by those blasted all-seeing eyes! Luthor: Perhaps. Perhaps not. Still — if we were free tomorrow, would you care to join forces? To make a small trade? Doc Ock: Hah! "My enemy for yours"? Why not, Luthor?
Doctor Strange's enemies Dormammu and Umar are a brother/sister pair of Dimension Lords. Usually they don't cooperate, which is good because it goes badly for him when they do. (Subverted in nearly all instances, since Umar is more clever, though less powerful, than her brother and wants to see Dormammu fall more than she wants to defeat Strange.)
Very quickly subverted in a Justice League of America story. Green Lantern calls into JLA and says that seven supervillains are trying to kidnap the President at the White House and that he and The Flash need support. The others assume it is this trope, but Green Lantern quickly cuts them off: they all got the idea independently at the same time, and are currently trying to kill each other for "stealing my idea!" He and Flash need help saving civilians from their crossfire, the president is safe. This turns out to be one of many signs someone has been misusing powers of probability manipulation.
Astro City doesn't tend to focus on supervillians enough for Villain Team Ups to be the focus of stories, but there have been a few groups that have shown up as peripheral details:
The Unholy Alliance is a recurring team of villains who team up for various reasons, though the members also work on their own. The roster changes a bit from story to story, but core members tend to be Demolitia (team lead), Slamburger, Glowworm, and Flamethrower.
In the "Tarnished Angel" story arc, Steeljack has a brief flashback to when he was part of the Terrifying Three — Cutlass, Steeljack, and the first Quarrel.
Steeljack: "We didn't last, and fought each other more'n' anyone else — but we were friends, I guess."
In Empowered, an animal-themed group of supervillains seek out the fire elemental Willy Pete for a teamup. Unfortunately for them, it turns out that Willy Pete's definition of "teamup" means he rapes and eats the people he's teaming up with.
Happens in Paperinik New Adventures: in issue #30, "Phase Two", it was discovered that Two had taken control of the Evronian computers in order to manipulate them to destroy his arch-enemy One. He is discovered but, since his plan will destroy Paperinik too, the Highest Zotnam chooses to form an alliance with him and let him work. Of course, when he fails, he decides to kill him.
A Simpsons comic story has a team-up between Sideshow Bob and the evil winemakers from the episode "The Crepes of Wrath".
When The Joker travelled to the Dreddverse in one of the Batman / Judge Dredd crossover comics, he joined the Dark Judges, a group of four undead lawmen dedicated to annihilating all life. After Joker convinces Judge Death that he could be useful and asks for immortality, Death placed his spirit inside a dead body to make him one of them. Eventually though, Joker tires of the arrangement because he found the constant mass slaughter to be getting monotonous.
The image above came from Spider-Man and Batman: Disordered Minds, where Carnage, a fanboy of the Joker, teams up with the clown in order to terrorize the world. However, the team-up quickly deteriorates as the Joker detests Carnage's straight-to-the-point murder and Carnage cares not for Joker's theatrics. The sequel, Batman and Spider-Man: New Age Dawning, features Ra's Al-Ghul forcing The Kingpin into an alliance by infecting Fisk's wife, Vanessa, with cancer without his knowledge and offering a cure.
The Transformers: More Than Meets the Eye: When they find out that Megatron has defected Tarn and the Decepticon Justice Division hunt down Deathsaurus and his crew and offer to stop hunting them if they team-up to hunt and kill Megatron. When the plan comes to fruition and they have their enemy in sight, one of the old Big BadsOverlord shows up, wanting to kill their prey himself. After a scuffle he agrees to join Tarn and Deathsaurus in their little mission.