Ah, that makes sense. Yes, everything else is working. However, pulling the fuse for the rain sensor broke the functionality of the one-touch (automatic down) switches. Maybe related?

For anyone interested, I found what looks like an official BMW document that's extremely helpful. It explains functionality of the ZKE III body electronics system, including the GM, and has an entire section on wipers and the rain sensor. I now believe I have a combination of problems That is to say, I have partial rain sensor functionality and partial standard intermittent (no rain sensor) functionality.

http://www.bmwmotorsports.org/BMW_docs/x5zkeIII.pdf

A few things I noted from it:

- The wiper relay is a double relay. If one stage of the relay fails, the wipers would still function, but be limited. This is apparently a common problem on E38s and E39s, which share the GM III system. I think I'll replace it after all.

- "When the wiper motor park contacts signal the GM of the wiper arm position, the signal is simultaneously sent to the rain sensor as an indication that the windshield has been cleared of water drops and causes the rain sensor to reset the sensitivity delay timer back to 0." With my wipers swiping twice, this means that something is probably wrong with the wiper motor park contacts, or at least the signal output by them. That means replacing the wiper motor.

- Vehicles equipped with a rain sensor do NOT use the vehicle speed to control wiper speed. Only the standard (manual) intermittent function uses this. Yet my X clearly uses the vehicle speed to change wiper speed.

- When there's a fault in the rain sensor circuit, the wipers revert back to standard (manual) intermittent function, controlled by the knurled rotary dial on the stalk. Yet, I don't have standard intermittent functionality, as the rotary dial does nothing.