We count tilings of the rectangular grid, cylinder, and torus with arbitrary tile designs up to arbitrary symmetries of the square and rectangle, along with cyclic shifting of rows and columns, generalizing and classifying a a tiling problem first enumerated by M. C. Escher in May 1942. This provides a unifying framework for understanding a family of counting problems, expanding on the work by Ethier and Lee counting tilings of the torus by tiles of two colors.