The problem was not the snowy-haily stuff on the roads. It was my supposedly-waterproof gloves. After 2.5 miles they were caked in ice and soaked completely through, and my fingers were stiff and aching. There was no way I was going to make it another 4.5 miles, so I turned around. The gloves were so bad that on the way back, it was marginally better to ride without them.
No tango class for me today. Maybe instead I'll drive to a bike shop and get some better gloves.