It's not that it's L-R- but that it's second order.
Second order crossovers shift the phase 90 degrees in one direction on the low pass, and 90 degrees in the other direction on the high pass, at crossover. That makes the output a total of 180 degrees out of phase at crossover. To avoid cancellation, a phase reversal of one or the other driver is required.
The L-R crossover- due to it already having a 6dB attenuation at the crossover point on each driver- is especially noticeable as being "wrong" if the polarity is incorrect at crossover. Cancellation will result in a DEEP hole in the response, at the crossover point- even more than with a Butterworth or other crossover types with less damping at resonance.
HOWEVER- that ASSUMES TEXTBOOK FLAT RESPONSE from BOTH drivers, through the crossover point. If either driver is rolling off on its own- it's OWN phase shift will ADD to that of the crossover. For example- if the tweeter is also rolling off at 12dB/octave (a second order highpass rolloff, which is typical of all drivers below resonance), near the crossover point- then it adds to the second order of the crossover itself- making a total of FOURTH order highpass. That has a 180 degree phase shift at the crossover point! Combining that with a second order lowpass, will make the phase only 90 relative degrees out, low-to-high, no matter WHICH polarity you connect the drivers. Under those conditions- usually, you'd want both drivers in phase- just for better step response, usually- but which ever way sounds better, is better...
The same applies, to the woofer- if the woofer is rolling off on the upper end of its response near the crossover point- it can act as a first or second order lowpass in itself. That again changes the total slope/order of the lowpass- making it either act as a third or fourth order lowpass. That again changes the phase relationship with the tweeter- either another 45 or 90 degrees of phase shift, which has to be accounted for...
This is why, for one example, something like an EPI M150- despite only having ONE cap in the crossover (first order highpass on the tweeter, and NOTHING on the woofer)- actually acts like a second order lowpass (due to the woofer mechanical rolloff) and a third order highpass (due to the combination of the first order crossover plus second order tweeter mechanical rolloff)! This is why most of these early EPI speakers sound better, with the tweeter in reverse polarity...
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Regards,
Gordon.