Once upon a time, there was a French term, Chasse, which referred to a sidestepping action (because one foot moves away, and the other foot chases after it).
In Square Dancing, this became a figure called Sashay, which was like a sideways Do Sa Do. A separate folk-modification transformed the figures Allemande and Chasse into All Around the Left-Hand Lady and See Saw Your Taw, where "Your Taw" is a corruption of autour, the French word for "around" (as in Sidestep Around).
Sashay and See Saw are both counter-clockwise (i.e., left-shoulder passes), and they differ in that Sashay starts with the person next to you, and See Saw starts with the person you are facing. See Saw is also used as the mirror image of Do Sa Do, but that usage is comparatively recent.
If Sashay took you all the way around, then Half Sashay would take you only half-way, and you would have changed spots with your partner. This was so useful and popular with callers that the original Full Sashay is now a completely obsolete call. There are only a tiny handful of references to it in Modern Western Square Dance, buried in Burleson's Encyclopedia, like Allemande Left, Go Allemande "A"; With A Right And Left, And A Full Sashay. Such a figure obviously dates back to the era of rhyming patter, which hasn't been fashionable for at least a couple of decades.
Now, the name Half Sashay is commonly shortened to Sashay.
Calls like Ladies Center, Men Sashay involve a slightly different meaning of Sashay, close to the original "sidestep" idea.
The idea of Rollaway came later, and at first involved the Ladies doing the Rollaway while the Men just did the normal Half Sashay. It is still sometimes called Rollaway With A Half Sashay. Some people believe that Rollaway called when the dancers are circling left, for example, means that the Ladies do the Rollaway while the Men merely stand still, rather than being the same thing as Rollaway With A Half Sashay.