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.)

Draw a line on the screen from top to bottom. You interpreted "rotate away" as turning it around this axis, which is the same axis you used for the 90-degree clockwise turn. You end up with the screen right-side-up, just facing away from you. It's the same thing I intuitively did.

Now draw a line on the screen from left to right. "Rotate away" by turning the phone around this new axis - so the top of the phone moves away from you, and the bottom of the phone moves closer to you. You end up with the screen upside-down, and also facing away from you.

Can confirm. Followed instructions, got the camera facing the same in both cases (left, with phone upside down). I interpreted the 180 part as flipping the phone around the horizontal axis.

The 180 first part was right. Make sure you are rotating 90 degrees in the same direction both times from your frame of reference (clockwise looking from the top).