g4-g8 chess conundrum solutions

As you may have discovered, the main difficulty in reconstructing the board does not have to do with the chess position, but with the checkering of the board. In fact, if you remove all the chess symbols, there are still only two different ways to use the pieces to construct a properly checkered 8x8 board (up to rotations of the board). This was determined using Peter Esser's polyomino solver "mops".

Solution to Puzzle #1

Note that four of the pieces can be rotated 180 degrees, those named X, Z, and I, plus the 2x2 square "O". Including the chess symbols, only piece X is itself invariant under rotations. When any of the other pieces is rotated the board is still checkered properly, but the solution is no longer valid, because black is no longer checkmated and/or the chess position is no longer valid.

Solution to Puzzle #2

If I or O are rotated 180 degrees white is no longer checkmated. If Z is rotated 180 degrees the chess position is no longer valid (pawn on rank 8).