Is the number of legal chess positions odd or even? 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.
Odd if you count the initial position and Even if not.

Mirrored positions

Almost any position, including checkmates and stalemates can be flipped by switching colors.

Mirroring the chess board horizontally between 4 and 5 rows, you cannot find a position that colors can be switched and produce an illegal position. So for each position there exists a mirrored position, except the start position.

The challenge would be to find a position that cannot be mirrored.

Non Mirrored positions / Corresponding positions

There can exist positions where it cannot be mirrored due to the "turn that cannot be switched". For its this position, there would be a "corresponding position" that also cannot be mirrored.

e.g. 1. Nf3 Nf6 2. Ng1 Nd5 3. Nf3 Nf4 4. Ng1 Nd3+

its corresponding condition is:


Nf3 Nf6 2. Nd4 Ng8 3. Nf5 Nf6 4. Nd6+


Both cannot be mirrored.

Triangulation

Due to Triangulation, all positions after a pawn is pushed can be mirrored. Non mirrored positions can be constructed only with knight moves, moving only knights cannot produce Triangulation. 

Start position

The start position, before any moves, the turn cannot be switched so that black makes the first move.

Calculating positions from knight moves

To solve this for sure, we need to calculate the total positions that arise from only knight moves.