The Avengers/JLA relationship: the two second-mightiest hero teams of Earth 777 (both take a back seat to the Defenders in terms of power) have what might be termed a complicated relationship. The two groups have alternatively cooperated, clashed, and maintained a "friendly" rivalry throughout their history. Truth to tell, the rivalry part of the relationship has mostly existed on the Avengers side, though the occasional League hothead such as Green Arrow nursed grudges against the "hot-shot New Yorkers" as well.
The League has generally enjoyed much higher popularity world-wide, oweing to the presence of Superman, J'onn J'onzz, and Wonder Woman on the team. Despite the widespread adulation of the Man of Steel, the Avengers hold the edge in the US, mostly because of the original Captain America's role on the team; other reasons include the JL's long U.N. association (distrusted among conservative Americans), Wonder Woman's controversial positions on women's rights and religion, and the distinctly PR-unfriendly Batman.
In point of fact, the federal backing usually given the Avengers stems in part from the US government's distrust of the Justice League (even when they were the JL of A); the League's most prominent members included two foreign monarchs, an obvious alien (as opposed to Superman, and agents of a purported extraterrestrial paramilitary group (the Guardians of Oa). By contrast, the Avengers have mostly been dominated by home-grown Americans (even Thor is generally assumed to be American, or maybe Northern European, his Tales of Asgard thought of as mere fables). Despite the inclusion of the Black Panther, Scarlet Witch, and Quicksilver, most Americans think of the Avengers as "our" team. Possibly this is because even foreigners such as Black Panther have tended to downplay any political differences with the USA while on the team (T'Challa has rarely seen action with "Earth's Mightiest Heroes" since becoming more involved in Wakandan issues).
Tony Stark long ago drew up contingency plans to deal with many of the Leaguers (the "Ragnarok" Thor clone was originally meant to deal with a rogue Superman should the need ever arise), mostly done at the insistence of the Feds. The Dark Knight has anti-Avenger designs and weapons because he's Batman; he had the same prepared against his own teammates after all. Wonder Woman has never really gotten along with the chauvanistic and somewhat barbaric (by Amazonian standards) Thunder God, and tolerates Hercules only because her mother Hippolyta long ago forgave the demigod, who was not in his right mind when he sexually assaulted the Amazon Queen. Hawkeye usually rubs his relative youth and superior physique in Green Arrow's face, while the Emerald Archer has flaunted his money in Hawkeye's mug (at least during the times when GA had wealth to show off). Aquaman squared off against Captain America on several occasions during the Sea King's darker days following his emotional near-collapse. Hank Pym always felt intellectually threatened by Ray Palmer.
Of course, not all members of the teams have sour relations. The original Captain America was on excellent terms with Hawkman (Carter Hall), whom he fought alongside several times during WW II, and Wonder Woman (he knew her mother Hippolyta while "the Big One" was raging too); Red Tornado and the Vision had an obvious bond; the "fighting females" (WW, Hawkgirl/woman, and She-Hulk) always formed a tight buch when the two groups teamed up.
But relations were sometimes strained enough for the teams to work out formal rules of engagement governing which group had precedence and in what situations.
Of course, whenever the Defenders (whose membership included Black Adam, Dr. Strange, the Enchantress, Eradicator, Hulk, Lady Quark, Namor, the Silver Surfer, and Warrior at various times) just ignored all such rules whenever they showed up on the scene together...
The Contingent: When all other super teams fail...