Well, the doctor thinks it's due to compression of nerves, and she made a guess that it is the "carpal tunnel" nerves that are the culprit. The bad news is she wants me to stay off the bike for at least a week.

She wanted me to stay off for longer, but I told her I had a charity ride in a week and a half, and she said I could get back on only if the symptoms didn't get worse. She also gave me a wrist splint to wear and some Alleve to take to reduce any swelling.
So I dragged out the stationary bike tonight and rode it without the handlebars. That thing is seriously over-geared! Next week I'll be back to the doctor's office for my knees...
Tim