Welll.... I hooked the two transformers up for testing. I put a 25 watt light bulb on the output and ran 24 volts in. I figured if if glowed, the voltage was low and I'd measure it. If it turned into a flash bulb, then I'd know they weren't close enough cousins. So.... 24 volts in.... not even a glow. So, I tried 110V in, and, oh no! The magic white smoke escaped from the transformer that I was feeding. End of experiment. I'll keep my eye out and see if I can find an exact twin for the one that should be OK. Both of these came from the scrap pile, but the owner told me he knew one was good. The one that smoked was the unknown one.
Or, maybe I'll find a cheap pure sine wave inverter. Whichever comes first!