Unsound...Where did you (and a lot of other guys)get the idea that a first order (6 dB/oct) crossover has no phase shift? Actually the woofer and tweeter phase varies with frequency, but always differs by 90 degrees. The tweeter can be physically positioned so that at some particular frequency, usually the X/O frequency, the two drivers are in phase. As you get away from the X/O frequency the phase difference between drivers becomes unimportant because one or the other is attenuated. Inverting the polarity of one driver changes the phase discrepancy from plus90 to minus 90...take your choice.
Polarity inversion makes more sense for a second order crossover, where the phase discrepancy is 180 degrees. The selling point for fourth order, apart from the 24dB slopes, is that, with 360 degrees of phase shift the drivers are in phase. Fourth order is difficult and expensive to implement with a passive speaker level X/O, but quite easy for an electronic line level crossover.
Polarity inversion makes more sense for a second order crossover, where the phase discrepancy is 180 degrees. The selling point for fourth order, apart from the 24dB slopes, is that, with 360 degrees of phase shift the drivers are in phase. Fourth order is difficult and expensive to implement with a passive speaker level X/O, but quite easy for an electronic line level crossover.