I've found a couple more variations on "all around your left hand
lady,"though I think the ones Tom has cited cover what I have
most often read.
Watch the sequence that starts at about 0:31 in this 1951 film
produced by Bob Osgood:
https://squaredancehistory.com/items/show/654
It seems to me that both gents and ladies face pretty much toward
the center of the square the whole time. The gents may be doing
most of the traveling to get around the ladies while the ladies
move more nearly straight forward and back, but the ladies seem to
be doing a least a little sideways movement to give the gents a
shorter path. The ladies' sideways movement is most evident after
the cut to a different camera angle at 0:35.
In _American Square Dances of the West and Southwest_ (1949),
Lee Owens writes:
"All around your Left Hand Lady" is the fourth variety of the
Sashay. In this movement the gentlemen only encircle their
Corner Ladies, passing back to back with them, then facing
center and dancing in front of them back to place, keeping
their backs to the ladies as they encircle them. As the
gentlemen dance around the ladies, the ladies dance two steps
forward and pause until the gentlemen have "cleared" them and
then dance two steps backward to place as the gentlemen
complete their encircling movement around the ladies. The
figure requires eight beats of music and is danced between
Corners. It is usually followed by the same movement danced
between Partners to the call of "See-saw your pretty little
Taw," or, "Sashay 'round your pretty Baby," ...
Owens's description of the See-Saw/Sashay part similarly says,
"The gentlemen keep their backs to the ladies throughout ..."
I can't recall seeing this styling recommended anywhere else
besides Owens's books.
--Jim