Polya Enumeration Formula gives us 6 equivalence classes of 2-colorings of a square. But, in the Polya coloring, the following 2 colorings belong to 2 different equivalence classes:

    00
    01

and

    11
    10

(0 and 1 are the 2 colors.)

What is the theory that groups colorings like the above 2 into the same equivalence class? The reason to put the above 2 into the same class is that we obtain one from the other by a mapping between the colors (0->1 and 1->0). Is there a formula to obtain the number of equivalence classes of colorings with this constraint?