Simple problem that requires a complex solution.
Just when you thought this thread was done...
I went down to Home Depot yesterday to do some pricing, and I took a few moments to look at breakers. The breakers in my panel now are the equivalent of a Square D QO120CP (tried to find a decent picture, but couldn't). The closest tandem breaker I could find was a QOT2020CP (again, no decent pictures). The thingy (for lack of a better term) that holds the breaker in the panel, the one that's not connected to any power, is different on those two breakers. Will that particular tandem work, or do I need to go look again?