In MWSD, walk around (your corner) is effectively a “without hands” Right Hand Turn – it specifically does not have the “look at the other person” aspect.
This is not in itself a reason for not using the phrase, but avoiding confusion with closely allied dance forms that share dancers is probably a good idea.
Jeremy
26. Walk Around the Corner
Starting formations: Squared Set, Infacing Circle Of 8
Command examples:
Walk Around the Corner
Walk All Around Your Corner
4 Ladies Chain; Circle Left; Walk Around New Corner
All Around the Left Hand Lady
Walk All Around the Left Hand Lady; See Saw the Pretty Little Taw
Dance action: Dancers face their corners. Walking forward and around each other while keeping right shoulders adjacent, dancers return to their original position, with their backs toward their corner. [underlining mine]
Ending formation: Right and Left Grand Circle
Timing: 8
Styling: Men hold arms in natural dance position. Women use both hands on skirt, moving skirt forward and back to avoid opposite dancer. Dancers should maintain eye contact over their shoulders until their partners become visible. [underlining mine]
Comment: Square dancing is evolving towards a preference for "Walk Around the Corner" instead of "All Around The Left-Hand Lady", which requires the women to mentally translate the command to "All Around The Right-Hand Man".