I'm using something a little different than you guys (I do have a dremel and one with a flex attachment, but...)..
It looks like this and I chuck it in a lathe and put 220 wet and dry on it. Woodcraft has them, I got mine at rockler. They're a bit cumbersome for paper, but the rubber top layer on the drum has a lot of give, which makes 220 wet and dry leave a forgiving finish. The downside is that you have to approach with caution on one side of a razor and avoid getting a catch or the thing could literally just rip a chunk out of a razor (I did that long ago with a dremel and pinged a chunk out of a razor, junking it instantly).
Attachment 184216
Of course, you have to have a lathe to use something like this - it's a heavy stable platform to use it and it allows you to keep control of the razor with two hands. I chucked it in a drill chuck, and not a jaw chuck - which would just have all kinds of spinning metal parts that could catch and fling a razor.
(I have no idea what the "right" way to do any of this is - I could stand to go back and run the drum into the top corner of the razor near the shoulder in the first picture, I see there's a spot that I missed a bit)