Orbit count for simultaneous colorings

Suppose a finite group acts on two finite position sets $A$ and $B$. Color the first set using $r$ colors and the second using $s$ colors, with both colorings moved by the same group element. If $c_A(g)$ and $c_B(g)$ are the cycle counts of its two induced permutations, the number of simultaneous colorings fixed by $g$ is $r^{c_A(g)}{s}^{c_B(g)}$: each cycle must be monochromatic, and different cycles can be colored independently. Burnside's lemma therefore givesfor the number of orbits. When the two position actions are equivariantly identified, this simplifies to the usual coloring count with $rs$ combined colors. For example, a tetrahedron's face is paired with its opposite vertex, and this pairing commutes with every symmetry.