An orb-weaver, I think. Anyway, this was a spider wasp, family Pompilidae. A female will paralyze a spider, then go prepare a nest in soil amid rocks or other debris (or have a nest pre-prepared). She will then retrieve the hapless arachnid, take it to the nest and lay an egg on its abdomen. The egg will hatch, and the wasp larva eat the still-living spider.
She left the spider there and flew away, perhaps to make sure the nest was ready to go. I found it interesting that the spider was deposited in such a conspicous spot. Perhaps the whole drag-it-around-the-grill routine was an effort to find a conspicuous landmark, so that upon returning from the nest she would have no difficulty finding the spider.
I went inside for a while and when I returned, the spider was gone. I never saw the wasp again.
Unidentified spider wasp, Family Pompilidae.