So I reached over and scrubbed it a bit in the snow.
Anybody remember enough chemistry to know what happened next?
Yep. There was enough salt still there to melt the snow on my hand--but the heat to do that had to come from somewhere and the handiest source was me. My hand hurt so badly from the cold I was afraid I'd done it some injury, and I probably would have if I hadn't gotten to warm water at the sink very quickly thereafter.
A word to the wise...