If you ever wire a loudspeaker channel the wrong way around, it feels very weird as a lot (but not all for a stereo mix) of the sound will cancel with the other channel in the middle of the speakers and you can move your head through this dead zone.
I've never tried it with headphones, but I suppose even though destructive addition isn't going to happen it might affect things like intended spatial perception.
I understand, and there could very well be some psychoacoustic cancellation when stimulating with a matching tone. But any effect will be regardless of the phase of the signal, because phase cancellation is a physical effect of sound waves that occurs before they are converted to neural activity (where tinnitus originates from).
In simpler terms: tinnitus does not have phase, at least not in the same way a sound wave does.
Neutral activity itself definitely takes phase into account somehow, since you can hear the phase difference of two same-frequency tones being played independently into each ear, without any kind of sufficiently strong mechanical connection that would explain it.
I think though that the brain is also considering the spacial relationship of the sound to the listener. Hearing the sound from an external source rather than "inside your brain" does provide relief in my experience.
Funny enough lots of stereo widening magic relies on a physical property of sound not existing when it's in your head.