Whoa. I kept reading past that, and the third answer by Logan R. Kearsley is almost exactly what I said, but better. I should have kept scrolling the first time

Whoa. I kept reading past that, and the third answer by Logan R. Kearsley is almost exactly what I said, but better. I should have kept scrolling the first time I opened the link! It ends by saying that in general relativity, "rotation can be considered as existing only relative to a certain choice of coordinates after all", though, so there seems to be more to it.