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In a right-handed wave, only the center dancers are holding left hands, so for a Left Swing Thru only the center dancers do the first part. Then everybody can turn by the right for the second part.
But from a left-handed wave, everybody does the first part. Each end dancer is holding left hands with the adjacent center dancer, so they all can turn. This brings the original end dancers into the center, holding right hands with each other. Only these two dancers do the second part. The other dancers are done and simply wait for the center dancers to finish the second turn.
Note: For a Left Swing Thru from facing couples, dancers should start by stepping to a left-handed wave. Then everybody turns by the left, and the new centers turn by the right.