Thinking of a checkerboard I wanted to do something similar with quasiperiodic tilings.
The checkerboard is a square grid with two colors. A rotation by 90 degrees around a corner exchanges the colors, but else the board remains the same. The four-fold rotational symmetry is thus a twofold color symmetry. Similarly, we can have a twofold color symmetry for the quasiperiodic tiling. A rotation by 45 degrees then exchanges the color of the tiles. Thus we have to use two distinct colors for the squares and two distinct colors for the rhombs. The orientation of the tiles determines their color.
The result depends strongly on the colors we choose. An example:
Using greys, we get rather different pictures if we put strong contrast on the squares:
or on the rhombs: