Combinatorial enumeration of 2×2 ribbon patterns

An algorithm for creating repeating patterns from a single decorated square gives rise to an obvious combinatorial question: How many different patterns can be created, following the rules? Answers vary according to the definition of equivalence of patterns, and computer sorting programs can provide numerical answers. But algebraic techniques give insight into the answers and provide general formulas for similar problems. Group actions on signatures assigned to patterns can also determine which patterns have symmetry.