For those who didn't know (like me, until I studied the Help in response to this thread) you can rotate using the Move Selected Pixels tool by right-clicking in the selection. This also applies to the Move Selection tool.
I have often wished for exactly this capability but never thought to try the right-click approach. Suggestion: rename "Move Selected Pixels" to "Move/Rotate Selected Pixels" (and similar for Move Selection), as a signal that this is possible, to signal that the capability is there. I simply assumed it was not possible.