One could devise many different decision trees to accomplish the same purpose. In this case, we can identify any pattern by asking no more than 4 questions. The parenthetical remarks merely summarize the partial information obtained at each stage of the decision process. We could omit them.
Most people find higher order gyrations easier to see than lower order gyrations.
If you've found the order 6 gyration points in a 632, they form the corners of equilateral triangles. Then the order 2 gyration points sit in the middle of those squares. If you've found order 4 gyration points of just one sort in a 442, they form the corners of squares. Then the order 4 gyration points of the other sort lie in the middle of the squares with the order 2 gyration points at the midpoints of the sides. A 333 has gyration points of order 3 of three sorts.