Thanks, Wayne!
In the end, you were right. When I initially set this thing up, it still had a miswired power supply with 60hz noise. That means I tuned the horizontal sine coil with that same noise in it, but I failed to recheck it after I corrected the problem. Looked at it with the scope the other day, and sure enough the first hump of the waveform was was lower than the second one! I watched the screen as I tweaked for equal humps, and sure enough that coil being detuned will cause the ringing I was seeing. Now I have nice straight lines on the screen, just like I should.