Noise cancelling headphones don't reduce sound, it just reduces your perception of it, which is why it can be dangerous to your hearing if you don't monitor the situation. It short-circuits your's brain's mechanism that says "this is too loud," when you'd normally step away from the noise source or wear protection. (Which doesn't stop some retailers from making dubious claims about how these units "protect" your hearing.)

Is this right? My understanding of the 180 phase shift is that the nett effect is that zero energy is transmitted to the eardrum or whatever. If there was +100 units of energy in the wave, -100 are added so the result is 0. As such, it is not reliant on 'short-circuiting the brain's mechanism': it is purely physics.

I would agree, though, that this analysis assumes a 'point-receptor' which the ear isn't, so monitoring the situtation is certainly advisable.