Thanks! That did it. (And just before dark!) I think maybe it makes sense, too. After reading your post, I remembered that this switch used to be a three-way before I moved here. The previous owner tied the other end together. Hence the red wire?

I've now got Black-switch to Black-#2 (with the red #1 still tied in), and Red-switch to Black-#1, and it all works. Does this seem correct?