Thanks all for your suggestions which led me to the Home Depot where I found a thing called a "Grinding Disc with Type 27 Depressed Center". The depressed center allows for flush grinding of a part against the face. The dimensions are wrong for my mount (and I would never do grinding on a lathe or mill) but maybe that's what it's for. I think the mystery is solved.
I had one like that years ago, it was for holding a disc of abrasive paper in a hand held electric drill. It also came with a rubber backing pad to give the paper a bit of stiffness. i think these were popular before flap discs.