I like how the holonomy adds a new level of difficulty.
I first put the tiles in the right spots, then tried reorienting them by cycling the tiles in each quadrant, but I had some kind of parity issue: the sum of the orientation offsets was 2 mod 4, and this sum was invariant under the cycling I was doing. So I scrambled and put them in the right spots again, and fortunately the sum became 0 mod 4, so I was able to solve.
I'm curious if the puzzle is solvable from any board state (e.g. the solved state with one tile rotated 90°, or two tiles swapped).