Is the number of legal chess positions odd or even? Two positions are not the same if they differ in castling rights (i.e. whether K or R have actually moved) or en passant capability (i.e. whether the move can actually be made) or who has the move.
The number of times position has been repeated is not part of the definition of position for obvious reasons, as is the number of moves since last capture/pawn move.
I don’t have the answer myself. This is hard but doable with a bit of programming I think.
EDIT: The answer so far and the comments are clearly heading along the right lines, although not always correct yet: terrific work. I want to give multiple +1s! I will summarise the key points to help focus the work. Pairing mirrored positions is essential i.e. those which allow triangulation. I term positions which have no mirror image “vampires” :-) Any vampire (except for the starting position) must be immediate offspring of a vampire, so it’s maybe easiest to just count all of them. Someone touched on castling, which is very important. Can you see how to handle castling systematically?
En passant is less important but amazingly there are vampire en passant positions e.g.: r1bqkb1r/2pppppp/8/1pP5/8/8/PPPPP1PP/R1BQKB1R where White retains at least one castling right, Black has at least queenside castling rights and White can capture e.p.