When piston rings are fitted, let us assume 3 rings (2 compression & 1 scraper) they are fitted with the gaps with a 120 degree offset to avoid blow-by (where combustion gasses leak past the rings - an issue that can happen when the gaps are all in line).
As the temperature of the piston increases due to the combustion temperature, the gaps close up due to the expansion of the ring, The rings may rotate, but that rotation will be very slow - however the linear movement of the piston up and down will be much greater.
The scoring can be due to combustion particles getting trapped between the ring and the bore surface or the skirt of the piston coming into contact with the bore surface. Once this has started then it just gets worse over time.
The gudgeon pin (aka "wrist-pin") in the piston is located fractionally off-centre to make sure that the piston is always tilted as it moves so that it does not "wobble" during the stroke, this wobbling can cause a noise called "piston-slap".
Edit: Another source of particles that can start scoring is the oil itself - if it is not changed regularly or other contaminants have entered the engine / sump...