It is a bit of a tricky topic, but the basics of it tie in with my other thread about Polarity vs Phase: A systems that flips polarity, will invert the sign of any input signal. Positive becomes negative, and vice versa. Whether people are willing to accept this or not, the way to characterise such a system, is by shifting all PHASOR phases related to this system at all frequencies 180 degrees(!). This does not entail any time delay of any kind(!). The delay aspects of the system cannot be seen at one frequency, since if you only know the phase at that one frequency; all you can see is what the steady-state phase will be (and how at each particular frequency the steady-state output will LOOK LIKE a time delayed version of the input; phase delay is not group delay). Study complex theory and signal processing, and this should be apparent. The phasor phase is just an initial phase, initial offset, or however I can explain it in words, if the mathematics is too difficult.
But can you hear this or not?? I do not know. I tried a few listening tests once, and I could not hear any difference, but some people might be able to, and maybe you can train yourself to some degree. But it could be down to non-linearities in the loudspeakers themselves, and if your hearing is also non-linear in that it has some half-wave rectifier functionality in it, then sure, why not. There is certainly something 'uncomfortable' about the whole moving inwards vs moving outwards thing, but the system will have the same magnitude and delay aspects as a one that does not invert polarity, and so one should not expect any difference, if linearity is assumed in the loudspeaker/hearing chain.
And then you of course you have the issue with music being comprised of many inputs from different microphones and different situation.
It is probably best to look at a single driver first and try to define the polarity of this on its own. Is the 'normal' polarity the one that gives you zero phase somewhere in the pass-band, and the 'flipped' one the one that gives you 180 degree in the same frequency interval? And which way should it move when you apply a battery; you need to look at it both transient-wise and steady-state, and consider the behavior of minimum phase vs non-minimum phase systems. I will try and find the time to show this with an example.... (busy, busy). But this page has some nice insights
https://www.prosoundtraining.com/2016/11/14/meaningful-loudspeaker-phase-response/
And when you then finally consider a complete loudspeaker system with each driver having its own reponse and cross-over section, and the total output probably being a non-minimum phase all-pass system, then most people lose track of the whole phase thing. Some filter topologies such as Linkwitz-Riley has the same phase for each section as for the entire system (sometimes called phase-coherent, I believe), and so magnitudes can be added directly, but it is not a linear phase so transient behavior is off. The main point for most designers is probably to have a fairly flat magnitude on-axis, and for some filter topologies this just entails a sign change falling out of the mathematics for some of the filter sections, and one way to obtain that is to flip the wiring; if you don't then you don't get the response you are looking for.
I am not sure how important absolute phase is. I think the linearity of the phase, which relates to the temporal behavior of the system, is probably more important, but this is just based on having listened to the Kii3s once in two different filter settings. It is difficult to discuss these topics online, as the mathematics needs to be in focus before all else, and that is not easily digestible for the typical audience.