The thing is, stainless is hard, but not so hard that use, and more importantly cleaning attempts, can scratch it, ruining that beautiful, uniform appearance.
On our stove, between the two burners you see here, a typical dishwashing sponge with a scratchy side was used to remove some baked on food. Sadly, it left a clearly visible remnant of the effort: an area where the scratches ran up-and-down rather than side-to-side.
Since the finish is originally created by sanding, I thought it might be repaired in the same fashion. But what grit should I use? Originally I tried 400, but it was too fine, producing an almost polished finish. So I backed down to 320: perfect. I know the photography doesn't really show it, but the cleaning 'fingerprint' is gone, completely.
NOTE: When sanding, you must be careful to always move the sand paper in the same direction that the original "brushing" went.
Of course, if the scratches are deeper than the ones that 320 grit makes, then it will take a long time to sand down past them.
I did the actual sanding with a 3M sanding sponge that I now have dedicated to the purpose, but I suppose that any sandpaper would work.
(BTW, I used the sanding sponge to renew a stainless head sink too. It worked just as well there. In this case, the brushing was circular, around the inside of the sink, rather than linear.)