A random orbit sander will not leave the parallel marks and if you use say 400 or 600 grit any scratches would be too fine to be seen by the unaided human eye but that might produce a shine as well.
From my experience in woodworking, I would try buffing the back with a linen buffing wheel with perhaps a bit of white diamond polishing compound. If the scratches are too deep for the white diamond I would switch to a canvas wheel with Tripoli buffing compound. If the buffing leaves the surface too shiny I would "break" the shine with 320 grit sandpaper on a random orbit sander. That will leave a very slightly roughened surface but the random orbit sander does not leave any parallel or straight sanding lines.
Random orbit sanders can be found anywhere woodworking tools are sold. The ultra fine grit (320, 400, or 600) sandpapers are harder to find, but any store catering to woodworking such as Rockler's or Woodcraft will have sandpaper in those grits.
