I am looking for retrograde and inverted remakes of classical pieces. Even better if they are being played simultaneously with the origina, like Pachelbel's Canon, Original Version and Retrograde Inversion Played Simultaneously.
The earliest version I can think of is Bach's Musikalisches Opfer which is a series of canons Bach composed after given a theme by the King. The music can be played forwards and backwards simultaneously harmoniously in various ways (refer to the video).
I also remember reading a score of Mozart's violin duet piece called "The Mirror". To play this piece two violinists stand across from each other and play the same score from beginning to the end from their own angles.
In simplest words, for this kind of music to work the (roots of the) chords in the piece must be symmetrical, i.e. the first note(s) and last note(s) must be in the same chord, so that when they are played at the same time it will sound natural. Rhythm would be another key factor.
As @Menglan pointed out, the crab canon from Bach's musical offering is probably the most famous example. I transcribed it and put it on musecore:
The art of fuge (Also Bach) has 2 fugues which can be inverted. Look for Contrapunctus XII and XIII (there's a rectus and an inversus version for both of them)
I wrote some mirror canons myself. Of course, I'm nowhere close to the grand wig himself, but I had a lot of fun writing these:
Both these canons are written so that you can put the page upside down and get the same piece again. And because of this property, you can play each piece in inverse too. So playing the rectus and the inversus right after each other yields a palindrome.