I'm sure that rings rotate - don't have an opinion on how fast, why, or if it is all rings (there are lots of types). I was told during my first motorcycle engine rebuild in 1968 to put the top ring at 10, the middle at 2 and the oil at 6 (oil to the rear). I've done that on every engine since no matter how many cylinders. The few that I've taken apart that I built always had the rings different from my setup and surprisingly usually still spaced about 120 degrees apart.
You would think that logic would dictate that cross-hatching, if it caused them to move at all, would make them move back and forth.
IMHO and only guessing: lining up the ring gaps should not cause more oil-related carbon on top of the pistons. The oil-control rings should still do their job. If related to the rings, I would say, gaps wrong, ring(s) upside down, bad hone, wrong rings for the pistons (groove fit), etc. Other possible causes: if oil tank and breathes into air cleaner, too much oil entering carbs, too rich (not oil carbon, gas carbon), ale guides leaking oil, blocked crankcase breather, etc.