> "The low pass section is inverted, since 12dB crossovers always invert the phase of one signal. If desired, the high pass section may be inverted instead."
Not the "phase of one signal", that's a mis-statement. BOTH filters shift, and the 2nd-order case comes out OUT-of-phase between outputs, so you flip one output.
Consider the first-order crossover, and just one side, say the low-pass. At DC, output equals input. At infinite frequency, output is falling 6dB/octave, and is 90 deg out of phase with the input. There is a frequency we like to mention where response is 3dB down from DC. At that point the phase is shifted 45 degrees.
Use a 1st-order low-pass and high-pass together, both with the same -3dB frequency. At crossover, both are 45 degrees out of phase from the input. The drivers are fed 90 degrees out of phase with each other. This "could" produce a little cancellation at crossover, but in real life never enough to dominate the design.
Use a set of 2nd-order filters, and now each has a 90 deg phase shift at the -3dB frequency. (True for Butterworth; Bessel a little less, Chevyshev a little more, but still near 90 deg.) The two drivers are fed 180 degrees OUT of phase. If you actually had them working equally well at crossover frequency before you put in the filters, then the filters would give a perfect cancellation at crossover, and in real life a deep dip is possible.
Of course the drivers are also filters and are rarely at the same effective distance from the listener. Sometimes it all works out so the drivers crossover best when "in" phase, sometimes best when one is inverted. So on the 12dB crossovers you do want a phase-flop switch. It probably does not matter which side it is on: absolute phase is 99% inaudible, and for those few musical settings and recording chains where it is audible, it will be "wrong" about 50% of the time. If you care that much, put phase-floppers on all Xover outputs and fiddle till happy.
When you get to 4th order crossovers, ideally the outputs sum in-phase, and in practice any notch at crossover is so narrow that the null won't be noticed in concert. In precision monitoring, it is an issue to struggle with.
> 7th-order elliptical lowpass
And when mixing 2nd-order with 4th or 7th order, especially non-Butter shapes, phase WILL get bent and there may be no "perfect" answer.
Eliptical makes little sense to me to cut the squawk out of a sub. Your ear response is rising fast. Subwoofers are sometimes too-effective midrange speakers. Any filter that goes down and bounces up is going to leak audibly, unless the bump-up is 90dB down or more than 4 octaves up. You want a filter that falls and keeps falling. Get Jung's Filter Cookbook. A 3rd-order 110Hz Butterworth and a 3rd-order 0.5dB 150Hz Chevyshev in cascade will be 10dB down at 150, 45dB down at 300, 80dB down at 600, and keeps going down down down.
Don't overlook acoustic filtering. Running the sub output into a fuzz-lined chamber with a selected output gives a 12dB/oct cutoff, not only on the input signal but also on any squeaks and slappings that the woofer makes. Just facing the sub into a carpeted corner gives a sloppy ~6dB droop.
> 2nd-order 80Hz highpasses on the five channels and buffer the highpass inputs into a summing amp into a 4th-order 80Hz lowpass for the sub
Ah. Then you have additional phase-shifts. 80Hz is 14 feet. So a 7 foot path difference between the sub and one main is a 180 deg phase shift; just 3.5 ft is a 90 deg shift. You probably can't put 5 mains and a sub and your head all inside a <<7-foot sphere, so you are loaded with phase shifts, different for each speaker. In general, all your mains go where they have to go and "must" stay in phase with each other. Then throw your sub where-ever, put a phase-flip switch on it, and try both ways while listening to a well-known walking bass line. It may make no difference at all (the sub can't cancel the bottom-note of ALL the mains at varying distances from the sub and your head). If you do find 80Hz (the note ~an octave up from the bottom of a bass) go weak one way and strong the other way, you know what is best.