Left <-> right interchange doesn't really happen anymore since radial tires have become standard on most vehicles.
Radial tires are unidirectional, so if you go left -> right your tires will be rotating backwards, and then they won't be able to channel water out from the treads.
Change the wheel a given tire is attached to. This is to have a regular "wear" on your tires. Incidentally, you can also regularly rotate your mattress to have a better sleep.
I don't think I've ever had this done when getting my tires rotated. They just move the wheels and tires from front to back and sometimes from left to right.
Allows for uniform wear and tear for the tires.
Some even interchange the left and right tires.