There is a version of pivots with man's leftward (lady's rightward) sway, usually with a promenade shape. Common entries are (1) Inverted Tipple Chasse (man forward, lady back), or (2) the Outside Spin variation where the man sways to L on 2 and opens the lady to a promenade shape.
The Running Spin Turn is also often accompanied by leftward sway on steps 3-4. The sway actually begins to kick in slightly earlier, between steps 2-3.
As for a pivoting type of action with rightward sway, you may have heard of something called a Lunge Roll. It's probably most common in Tango (preceding the Stop Lock), but versions exist in Waltz, Foxtrot and Quickstep, too. Technically it's not a rightward sway that's held throughout the entire turn; You generally transfer weight to the RF (LF for lady), then commence sway. And mind you, this is not just any normal sway, a good lady will really lay this one out (hence the "Lunge"), pivoting all the while.
And here's another one: The Overturned Spin Turn to Turning Lock to R. It's not just a spin with straight rightward sway, however. Like the Lunge Roll, the sway starts kicking in mid-movement. Think of it as a smooth Lunge Roll with rise... The body is straight as you approach the RF, then sway begins once the weight begins transferring, and continues to increase as you move away from it and towards the next step. The rightward sway is then held through the beginning of the Turning Lock, straightening out between steps 2 & 3.
Regards,
Jonathan