Is that a series string filament set with no power xfmer? The fact that it is a single hum bar and not two means that the hum rate is at 60 Hz while the vertical scan rate (off air color signal) is at 59.94 Hz. Hence the slow roll. A heater to cathode leak/short in one of the tubes in the video chain or CRT would give you two hum bars I would think. A single bar I'm thinking may be power supply hum from a half wave voltage doubler supply. Bad electrolytic? I've none to little experience with series sets so maybe one of the experts here can chime in.
The fact that it inverts with fine tuning adjustment...
That sounds like the hum bar is in the RF signal. Try replacing the RF amplifier tube if you have one in the tuner. Check the RF signal source with another TV. Maybe your cable company is sending you this hum bar.