Fascinating. But would this effect be constant across all recordings with different absolute polarities?Yes, this has nothing to do with absolute phase . . . it's more of a component-layout issue.
Here's an example - if you were to take two identical raw crossover inductors, and wire them in series . . . the electrical result is theoretically double the value of a single one, that is, the values add. However, if you stack them one on top of each other like doughnuts, the two magnetic fields around them will interact, and the overall inductance will either increase or decrease, depending on which way they're stacked. Because stacked one way, the fields will be going with each other and combine, but if you flip one of them over the fields will be working against each other thus cancel each other out a little bit.
But of course instead of physically flipping one of them over, you could simply reverse it's leads and get the same result. Now in your case, my speculation is that the interaction is between the woofer and one or more of its associated low-pass series inductors. Here, reversing the relative phase between the crossover and the woofer could somewhat change some of the inductors' effective values, and this would happen even if the polarity was also inverted on the speaker input to preserve absolute phase.
One of the downsides of using a steep sixth-order crossover design is that the required tolerances for the component values is much more critical to acheive the desired crossover slope. And changing the crossover slope affects not simply the summed frequency response, but also the speaker's directivity charactericts through the transition band - and this is something I would very much associate with a perceived change in imaging.
