I was playing around with a 2x2, and discovered that these two sequences are identical:
R U' R' U2 R' U R2
and
R2 U R' U2 R' U' R
R2 U R' U2 R' U' R
It's even funnier when you notice that the order of the moves are reversed, but not the moves themselves.
I also know from before that on a 3x3 the following two sequences are equal:
R U R' F' U2
and
U2 F' L' U L
R U R' F' U2
U2 F' L' U L
I suspect this is one of the shortest such pair. Obviously it works on 2x2 too, but is there any shorter for 2x2? What other interesting, non-trivially equal sequences out there for 2x2 and 3x3?