I had the idea of making a Truchet tiling holder that lets me rotate each piece by 90°. There's a little pin in the tray and a slot under each piece that allows each insert to rotate and line up in one of two orientations.

I used Random.org to choose the tile orientations for this tiling:
11010111
11001111
00111100
11111111
11100110
01101111
00000101
00101011