If you start with the phone upright and rotate the screen away from you by turning the phone around the vertical axis, then both rotations are around the same axis and of course they do commute.
My guess is that romwell is holding the phone flat, so that the rotation away from you is about a horizontal axis; then you should experience the noncommutativity.
(The resulting orientations are 180 degrees apart, which indeed makes it difficult to say that any one orientation should be the unique average. But this is due to the geodesic structure of the space of rotations, not the noncommutative product that happened to construct these points, see above.)
My guess is that romwell is holding the phone flat, so that the rotation away from you is about a horizontal axis; then you should experience the noncommutativity.
(The resulting orientations are 180 degrees apart, which indeed makes it difficult to say that any one orientation should be the unique average. But this is due to the geodesic structure of the space of rotations, not the noncommutative product that happened to construct these points, see above.)