The Freedom League and the Patriots
All around the world, a lot of people want to ask whether or not the Freedom League could beat up the Patriots. Then they get into arguments over which members of the Freedom League should be present and at what era amongst other internet pedantry. Thanks to the wonders of mind control, the two teams have actually tussled in the past with the results being inconclusive as to which is the stronger (though Divinos nearly defeated the entire Freedom League once when particularly angry---he was eventually taken out by Raven's quick use of a Annihilation Zone arrow given to Bowman).
In general, the two teams don't worry themselves about such things. Neither team really cares about the fame that is so important to others and is mostly concerned about saving lives. In that respect, they've often teamed up against particularly cosmic threats with little difficulty (other than who is in charge.)
Unfortunately, while the two teams are friendly to one another, they're not always polite. Aside from Johnny Rocket who can visit, have a cheeseburger, and be back in Freedom City in seconds; the two teams don't get to interact much outside of business. The merged universe is filled with a cornucopia of hostile, super villains, and terrorists that precludes super hero social parties. The loss of the Centurion and the retirement of many "Silver Age" (1980s for the Freedomverse) super heroes from the Patriots has resulted in the two teams having a lot of members neither side knows very well.
Ironically, individual friendships between the teams are actually quite strong. It's just that they don't extend on a group level.
The closest relationship is Athena and Daedalus. The Goddess of Wisdom and Daedalus have known each other far longer than their respective teammates. When most of said fellow teammates are dust and bones, they'll continue to know one another. Daedalus has a mixed relationship with the goddess, simultaneously reverent and disgusted at the same time. He's a little perplexed as to why she's allowing herself to be treated as just an exceptionally powerful mortal but likes to think that is a sign his time of ordeals is coming to an end (unfortunately-he's wrong. Athena's reasons for visiting Earth are unrelated to Daedalus).
Less close but still notable is Black Wing II's association with Raven. Or, more precisely, Callie Summer's relationship with Alexander Timons. In the merged universe, the original Black Wing took a great deal of inspiration from the Raven (just as the original Raven took a great deal of inspiration from the Golden Age Gray Wing). The two know each other from their civilian identities as well, in addition to their super hero ones (both deducing the other's identity fairly quickly thanks to their superior Detective powers).
In the pair's case, both are intimately familiar with the other's lifestyle. Just as Doctor Sin is Callie's grandfather, so is Death Mask III the father of Black Wing II. Likewise, they have somewhat dubious relationships with their father-figures (though Callie is at least on speaking terms with Duncan Summers). In the few periods where either has free time, they actually socialize at each other's estates. This has lead tabloids to add debutante Callie Summers to the list of Alexander's famous Hollywood girlfriends (Callie approves of its use for a potential cover but frequently chides Black Wing II for a way of life that she doesn't approve of).
Johnny Rocket's relationship with Splotch also bears mentioning. After a brief encounter with some of their super villains teaming up, the two struck up a friendship based on being the "Other Guys" in their teams (though that's since changed). Very similiar to the Human Torch-Spiderman and Booster Gold-Blue Beetle relationship, the two had a series of team ups in the first five or six years after the Centurion's death that were very high publicity. Their similiar ages and joking lifestyles made the pair's adventures quite light hearted in otherwise serious teams.
The relationship hit a rocky patch when Johnny's homosexuality became public knowledge. While Splotch was initially uncomfortable thanks to his Jewish upbringing, it was mostly an issue that there was an entire side to his friend that he didn't know. While there was some joking speculation in the media about their relationship, Splotch tried to mend fences with his friend after coming to terms with the discovery. Unfortunately, the two have never really recovered the trust that was lost between them.
In any case, the Freedom League's recent move to space hasn't helped the two teams much. Divinos, who lives in space, believes the Freedom League is taking on far more than they can handle by trying to protect the entire world while Captain Thunder believes the Patriots are unnecessarily America-centric (discounting that the Steel Commando is an operative for the United States government).