Good, but now for the kicker. There's a reason I didn't specify the order in which I pulled out the rugs in the OP.
After I finished pulling out every rug and walking the pattern forever, I went back and picked up all my rugs. Then instead of alternating odd-gons and 4x-gons, I placed two 4x-gons in a row before placing the next odd-gon. I still turned the same direction on every rug that I did before: still clockwise on every odd-gon and counterclockwise on every 4x-gon, so in the pattern 4-gon, 8-gon, 3-gon, 12-gon, 16-gon, 5-gon, 20-gon, 24-gon, 7-gon, 28-gon, 32-gon with clockwise turns on blue and counterclockwise on red.
After that second forever, which way was I facing?