I don't think I can completely agree with you that there is a dc shift if I apply negative sign to the complete waveform.
Oops, I mistyped and got it reversed. I meant to say that a phase change only would cause the offset, not the polarity flip.
Which goes to what I've been saying all along; when we all talk about phase inversion, we are really taling about polarity inversion, in which, yes, all the values are conserved because everything continues revolving around 0DC as the standard. That's cool. no problem.
The problem is that when explining this stuff to someone who is just wrapping their brains around it for the firts time, it's important to understand that what everybody calls "flipping the phase" is really more on a technical level than *just* flipping the phase. Whether it due to, as RD suggests (basically admitting the difference right there), removing offset as a factor by adjusting the centerline of the waveform's envelope to equal 0DC, or whether it's because of a DC rectification, or whether it's because in reality it's just a simple polarity inversion (which seems the simplest answer to me), there is actually more going on than an instantaneous ohase shift. Whether it's a polarity flip, phase plus DC reset, or any other formula, its NOT just simple phase change. That's all I'm saying.
Think about it from an entirely different direction. Imagine instead of a "phase invert" button on you mixer, you had a hypothetical "phase change" button instead that was accompanied by a knob that rotated from 1° to 359° (with a handy detent straight up at 180°

). We'll pretend that this was a circuit that allowed you to dial in an instantaneous phase change of any degree you wish rather than just a single choice of 180° only.
Such a circuit would not be able to use a polarity flip to do it's thang. Why? because phase and polarity are two different things. Such a circuit would also have to execute a phase change "in place" - instantaneously "shift" the phase so that the waveform rotates around itself and not around 0DC. This hypothetical circuit would have to incorporate an extra stage or two besides just the phase shift in order to keep the energy balance around the 0DC the same; i.e. to make sure that there are no unwanted side effects outside of the phase change itself. Why? Because once we talke polarity out of the mechanism and look at just phase alone, the 0DC balance issue is no longer automatically nulled.
The whole subject just screams that phase shift and polarity shift are not the same thing. I just simply don't understand what is so heretical or blastphemous about saying that, because it is - by every way I can slice or dice it, anyway - the truth.
And if it is true that phase change and polarity change are indeed different processes that in and of themselves (i.e. by definition) yield different results, which has been shown in this thread several different ways to Sunday, and if the inverter switches on your microphones and our mixers are actually performing polarity inversions (or possibly phase changes with extra offset correction to make it appear exactly like a polarity inversion), then it is perfectly true from a technical standpoint that the label "phase inversion" is a misnomer.
On a day-to-day level we don't really care about the historically esconsed misnomer because we all know intuitively that it really means polarity inversion. But when trying to describe to someone from another planet what's going on, the phrase "phase inversion" is misleading because executing a true, pure, instantaneous phase inversion only is different than what the switch actually does.
I really REALLY don't understand what is so difficult about getting that point across. You guys are smart guys. I can't explain it any other way than to say that I am battling against such deeply ingrained prejudices that I sound heretical: "How DARE you say that Rupert Neve is human enough to accept a commonly used name for something that is technically not quite accurate?Rupert would never do anything inaccurate because he is an engineering MACHINE!"
Here's how: Rupert uses the nomenclature of the business, his mixers are tools for seasoned engineers, not training modules for newbs. If everybody finds it easy to call it the old-fashoned and intrenched "phase inversion" instead of calling it "modified phase inversion via the flipping of polarity", who is ol' Rupert to argue? Everybody knows what it *really* means anyway.
Until a couple of generations later, that is.
G.