[vulcan01]

I think you can drag on the space around the cube to reorient without making a move.

[gregfjohnson]

(OP) You're right - left-mousing outside of the cube and moving the mouse around re-orients the cube. Left-mousing a face and dragging across one of the four boundaries of that face will rotate the face in the direction of the boundary you crossed.

I'm wondering if there would be a clean way to communicate the above to the user other than having them experiment and find it on their own.

[vulcan01]

You could try putting double arrows around the corners and sides. Like these (unicode): ⬌ ⬍ ⤡ ⤢