Schattschneider's problem on classifying patterns

Schattschneider's problem can first be stated for a tiling of squares. This is symmetric under the group generated by translations $T_1(x,y)=(x+1,y)$, $T_2(x,y)=(x,y+1)$, the rotation by $90^\circ$ given by $R(x,y)=(y,-x)$ and the reflection $W(x,y)=(x,-y)$.

Suppose we have a collection of four figures $A$, $B$, $C$ and $D$ which are pictures to place on a square. How many ways can we assign one of these four to all the squares in the tiling in such a way that the assignment is invariant under the translations generated by $T_1^2(x,y)=(x+2,y)$ and $T_2^2(x,y)=(x,y+2)$? We say two assignments are the same if we can apply one of the symmetries of the square tiling to one assignment and obtain the other.

This is trickier than it sounds, because it depends on the relations between the four figures, i.e. whether or not some of them are rotations or reflections of the other.