I suppose, if the game doesn't end immediately upon solution, of course it could be possible to have both an even- and an odd-length solution: if it's possible

I suppose, if the game doesn't end immediately upon solution, of course it could be possible to have both an even- and an odd-length solution: if it's possible to solve ending on a white square, then it's also possible to give a longer solution consisting of the first solution followed by winding around hitting every square on the board once more.