Title tells it all. The Stones did Otis Redding’s “That’s How Strong My Love Is”, and Otis answered a short time later with his famous cover of “Satisfaction”.
Bob Dylan, early in his career, covered Elvis Presley’s “That’s All Right, Mama”. ELvis returned the favor with his version of “Tomorrow Is A Long Time”. Dylan was very proud of that cover.
Steve Earle and the Supersuckers made a record together where they covered each others songs. Does that count?
(That record also featured Steve with the Supersuckers covering the Sones’ Walk Before They Make Me Run. Not what you were looking for, but I want to point it out because it’s a goodie.)
Looks like Steve released one of those on one of his own records:
I know that album, it doesn’t really count because it’s a collaboration, essentially Steve Earle with the Supersuckers as his backup band. But great album!
Well, “That’s All Right, Mama” also wasn’t an Elvis original but a Arthur “Big Boy” Crudup song, but I think in both cases the cited versions were the crucial ones.
Probably not quite what you’re looking for, but when I took Mrs. solost to see Sting and Peter Gabriel on their Rock Paper Scissors tour in 2016 (jeez, I think the last time I was at a concert) they covered a couple of each other’s songs.
Speaking of Peter Gabriel, back around 2010 he was putting together a project called Scratch My Back/I’ll Scratch Yours, which was intended to be a pair of albums, one where Gabriel did covers of various other artists’ songs, and the other where they would do covers of his songs. Unfortunately several of the other artists dropped out of the project and never got around to doing their Gabriel covers. I think both albums were eventually released but it wasn’t quite the way Gabriel had originally envisioned it.
At Desert Trip in 2016, the Rolling Stones did an impressive cover of Come Together. The next night, Paul McCartney reciprocated by performing a “Stone’s song”…I Wanna Be Your Man.
They wrote it for the Stones who recorded it first so the Beatles technically covered it. McCartney could have picked any number of Jagger/Richards tunes, but he kinda pulled a dick move by choosing a Stones song that he himself wrote.
Elton John covered “Pinball Wizard” for the film of Tommy, and slipped a bit of “I Can’t Explain” into the outeo on the soundtrack album. 17 years later, the Who reciprocated by covering “Saturday Night’s Alright For Fighting” with “Take Me To the Pilot” in the outro.
In live performances when they toured together, Elton John played “I Love You Just the Way You Are” and Billy Joel played “Your Song” to open the concert.
I’m sure I read some critic or other claiming that Dylan’s version of The Boxer was revenge for what S&G did to Times. I guess it’s plausible. But (if you accept that it was never intended to be serious) I rather like Dylan doing The Boxer.
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ETA - I know you know their work far better than I do, so: did the Stones ever cover a Gram Parsons song?
That’s mighty sweet of you, and I’m embarrassed at the same time I didn’t think of it myself.
None that I know of, nothing officially at least. Maybe somewhere there’s a bootleg cassette with home recordings that has Keith strumming and singing a Gram Parsons song, but I’ve never heard of any.
ETA: for the uninitiated, what @Treppenwitz is alluding to: Gram Parsons and Keith Richards were big buddies. The Stones recorded “White Horses” in 1969 at a legendary session at Muscle Shoals that spawned the nucleus of what was to become “Sticky Fingers”, but the final recordings and release of the album got delayed by label politics (they changed labels) until 1971. So the Flying Burrito Brothers released their cover of “Wild Horses” a good half year before the Stones in 1970 on their second album “Burrito Deluxe”, with the approval of the Stones, a rare case of a cover released before the original recording.
Elvis Costello covered “A Good Year for the Roses,” first recorded (but not written) by George Jones.
George Jones recorded an album of duets with other artists. On it, he and Elvis Costello sing “Stranger in the House,” which was written and first recorded by Costello.
I wonder if there are some Motown acts who covered each other? Certainly a fair number of Motown hits were passed from group to group.