26.? is an impossible puzzle even if the players spent the first moves moving the knights forth and back: the move number would be odd."

You've actually raised a very interesting question:

What if the knights switched sides -- such that on the 26th move, the knight at b1 is actually the king's knight, and the knight at g1 is the queen's knight, and they had together had spent the whole game maneuvering around to end up in each other starting squares? If they had done this, could it even be technically possible for this to be the position with white to move on an EVEN move, such as move 26 (or any other even-numbered move)?

Also note: while the knights are out moving around, the rooks would both be free to move back and forth from a1 to b1 and from h1 to g1 -- but they also therefore could not lose a tempo either, since to restore the starting position each rook necessarily would have to make an even number of moves to get back to rook one.

Furthermore, even if each individual knight would necessarily "lose a tempo" to get to a square of the opposite color, since BOTH knights must do the same dance, the second knight would also always lose a tempo as well, so as a group of two, the knights can NOT lose a tempo.

Thus, on general principle, it would seem that the answer is "No" -- even considering the possibility of switching knights, and the possibility of shuffling the rooks, under no circumstance could the starting position of a chess game ever be white to play an even-numbered move!

Pretty amazing, actually.

Pretty easy and obvious for a Friday puzzle.>

I fail to see what are the options after 1...c5, possibly your analysis is incomplete.

You aren't.

Although both knights might not follow symmetric paths (for example, Nc3-e4-g5-h3-g1 and Nf3-e5-c4-h3-b1) the essential point is that knights move alternating square colors with steps of fixed length. Therefore, they cannot lose tempos and since there are two the maneuvers to go out and return home involve necessarily an even number of moves.