Looking back over the planar symmetry groups, I noticed:

1. The 4-tile blocks can be viewed as two pairs of tiles on top of each other, beside each other, or diagonally from each other.
2. The two tiles in any given pair can be identical, or they can be different.
3. Except for two instances (Groups p4 and p4gm), the tiles are either right side up or upside down, but never sideways.