I brought my car into the shop today since it was making a scrapping/grinding noise when making a left turn, sounding a bit like an old van. Turns out that the problem was that the left front rotor was hitting the backing plate (which seems to have the function to hold the parts, but not the caliper, of the disk brake in place, the rotor being the disk part of the disk brake -- ah, it may mostly be the hand-brake drum brake), so for $64.78 the backing plate was adjusted to no longer do this. No sign of any damage to the rotor that I was told about, but we'll see. The part of the backing plate that I think was scrapping against the rotor doesn't look substantial to me and it is unclear to me what its function is, besides being a sort of mud catcher for the rotor.
I suspect the cause of this was the alignment I got when I bought new tires back in early June, something that shouldn't happen. So count this against Town Fair Tire in Hamden.
Here's a backing plate behind a rotor on a disk brake (in this case an E39 BMW, also see some nice photos of cutting back the baking plate so it won't scrape the larger rotor for this brake upgrade):
Update #1: having felt the backing plate with my hand, this piece of metal is so insubstantial that there is no way it is going to damage the rotor. The puzzle now is how the backing plate keeps its shape as well as it does in order to not scrape the rotor and make noise all the time. Not too many flying rocks under foot it seems.