Circular checker with nothing on it: 1 orientation state. With a line segment drawn on its diameter and rotation increments of 45 degrees: 4 states. With an arrowhead at one end of the line segment: 8 states. If it can be flipped (with distinct sides, e.g., color) with the an arrow also on the other side: 16 states. Automorphism group of an octagonal prism, dihedral group of an octagon.
This can be done with a typical coin.
8 states without flip is more than enough to denote the 6 possible chess pieces per side. Pieces are simple checkers with arrows: the direction of the arrow signifies what kind of piece it is. The color of the checker gives its color. Perhaps have drawings of the 6 pieces around the edge of the board to help players remember which direction corresponds to which piece. It is easy to cheat, or make a mistake, and change the identity of a piece by altering the angle of its arrowhead.
One can consider rotations of polyhedra other than the flat octagonal prism. A cube with distinct faces has 24 axis aligned orientations, or 48 if allowing 45 degree angles. This could be a use for the "joke" 1x1x1 Rubik's cube, a game with 48 different piece types.
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