Small nitpick, XOR is invertible when parameterized by either of its inputs, whereas OR and MOD are not in general. I.e., (x,y) -> (x,xor(x,y)) is invertible,

Small nitpick, XOR is invertible when parameterized by either of its inputs, whereas OR and MOD are not in general. I.e., (x,y) -> (x,xor(x,y)) is invertible, but (x,y) -> (x,or(x,y)) is not for arbitrary x,y and neither are (x,y) -> (x,mod(x,y)) or (x,y) -> (mod(x,y),y).