The Beatles and the Rolling Stones got along great behind the scenes. A lot of the rivalry was ginned up by their respective fan clubs and the press.
CC
Beatles vs. Rolling Stones....
I know once in a while one or another would drop in to encourage the others at recording sessions and were amiable.
One of the books dealing with this is Beatles vs. Stones
by John McMillian (2013). I have a few dozen others on rock in the 60s and both groups.
Brief synopsis says:
McMillian reveals how music managers helped to construct the Beatles-Stones rivalry as they set out to engineer moneymaking empires. He explores how the Beatles were marketed as cute and amiable, when in fact they came from hardscrabble backgrounds in Liverpool. By contrast, the Stones were cast as an edgy, dangerous group, even though they mostly hailed from the London suburbs. Although the Beatles always sold more records than the Stones, the Stones seemed to win greater credibility with the “right” types of fans: discerning bohemians, as opposed to hysterical teenyboppers. Later, the Beatles embraced Flower Power, while the Stones briefly aligned themselves with New Left militance. Ever since, writers and historians have associated the Beatles with the gauzy idealism of the “good” sixties and portrayed the Stones as representatives of the dangerous and nihilistic “bad” sixties. Beatles vs. Stones explodes that split.