Basicly this is my suggestion: (Note, not even as random as I had said, its almost exactly the same as we have now, but some of the faces on some of the configurations are rotated a bit)
So with only 4 configurations (which basicly match the 4 configurations we have “right now”, largely speaking we still have 2 vertical ones and 2 “sideways” ones), any of the 6 faces can have any rotation of the texture pattern. And you could use a tool to hit one of the 6 directions to rotate “that” direction 90 degrees, and even the current placement system when putting the blocks down would only require a very minor change to fit with the new orientations.
Here I had an entirely non-mirror P pattern texture, but for some of the cases atleast in the game, the texture has atleast one mirroring axis already, so 4 rotations actually does give you the 2 flips as well because of the texture already being mirrored (Eg imagine the P in this diagram is a T shape instead, or is a texture with just a diagonal line in one corner)