Hi tomthiel,
Yes, quite a few well regarded speakers over the years/decades have used/still use second order crossovers to the woofer. Given the nominal 180 degree phase delay, the most satisfactory results are generally gotten by inverting the relative phase of the wires to the woofer. This works well for some frequencies away from the fc but not so good for time alignment in the vicinity of fc. Here (with time shifts available) I’ve stood that on its head a bit by not inverting and delaying to seek best phase blending at fc.
I may be telling you a bit more than I know for sure, partly as discussion stimulation. (smile) It has been a few years since I looked more carefully at it. I just reviewed some old modeling spreadsheet work where I was showing myself that if I differently pro-rated or scaled the importance of the time alignment at frequencies stepping away from fc in half octave jumps, some of those schemes would start to suggest I might be better off with something headed toward an additional 35 degrees or so of phase shift. I thought that result was pretty small for how highly speculative I was being… and decided not to pursue it even as a listening experiment.
The overview message is still that, if one is willing to go there, DSP devices give us some greater freedoms to customize that integration than we’ve had in the world of analog crossovers.
Yours, John