First order Butterworth crossovers (6dB per octave roll-off) are favoured by many enthusiasts because they suffer the least insertion loss (3 to 6 dB in typical multiway crossovers) and are said to be phase coherent. In other words it is recommended that tweeters fed by a single cap should be wired in phase with the bass / midrange driver and all will be well. This is I think untrue since the use of a single cap results in a 90 degree voltage lag (in effect the capacitor has to charge up before discharging, but this is probably an oversimplification) such that it probably doesn't matter whether the tweeter is wired in phase or out of phase. Other people have pointed out that having a first order crossover results in a combing effect between the interaction between the two driver's outputs beyond the crossover point resulting in a series of reinforcements and cancellations. Some recommend for example that an additional supertweeter should face the rear (Snell E and the far-field version of the famous Cain and Cain Abbey) or point up to reduce serious combing effects. (This probably results in distributing the combing i.e. more but at a lower magnitude). My solution is to use second order Linkwitz Reily crossover to feed my supertweeters 180 degrees out of electrical phase (i.e. polarities reversed)  which I have convinced myself is better sounding. All very interesting but here is another perspective on the issue (http://customanalogue.com/elsinore/elsinore_17.htm) which argues that the electrical phase of say a tweeter is not the same thing as acoustic phase. For example the physical movement of say a tweeter dome turns positive (i.e. starts moving forward) immediately after the electrical phase is maximally negative i.e. at the bottom of the sine wave. I'd be interested in what members make of this argument. At issue I guess is the difference between electrical and acoustic phase.

See this is why I build other people's designs. I sort of follow where you are going but my head hurts... 😉

I am currently using a 1st Order 'Solen' split xover, with tweets opp. phase.

Still a work in progress.

DarkLantern blog - darklanternforowen.wordpress.com

Have you any further information of the Solen split Owen? I had a look and came up with this but re active crossovers.
books.google.co.nz/books?id=snlTDwAAQBAJ&pg=SA4-PA7&lpg=SA4-PA7&dq=solen+split+crossover&source=bl&ots=xCIal6hvi0&sig=ACfU3U18iBXflamR2zQua9PmvPWG8JI66g&hl=en&sa=X&ved=2ahUKEwiotJ7Et7_tAhWb4nMBHQnWBrQ4FBDoATACegQIARAC#v=onepage&q=solen%20split%20crossover&f=false
It suggest that whereas a Butterworth first order crossover aims to have both poles 3dB down at the crossover point and both speakers wired in phase, the Solen split suggests a 6dB drop (poles apart!)with one drive unit wired out of phase. I'm not sure whether in your application (we both use similar wide-banders) you use both the low pass and high pass sections or are using the high pass with a supertweeter only (mine comes in at 15khz). If it is just the high pass you are using, wouldn't using the Solen calculation simply mean pushing the crossover frequency a tad higher? I have the same problem (?) with my 2nd order Linkwitz Reilly but my reasoning is that my driver rolls off at 15khz (theoretically 6 dB per octave but as cones are far from pistonic more likely 12db or more per octave) I'd match it with a second order high pass.

I referred to Douglas Self's chapter on First Order Crossovers in his The Design of Active Crossovers.

DarkLantern blog - darklanternforowen.wordpress.com

When I read first order croosover, the late Jim Thiel comes to mind.

He was a leading proponent of first order crossovers as well as being aware of the drawbacks.

PS How could I forget Robin Marshall's wonderful Epos speakers ?

More head-hurting boffin stuff. I spotted on TM a reference (Listing #: 2902290045) to lattice crossovers in which it is claimed there are zero phase or time delays. Just like the Solen Split I have never heard of them but Wikipedia was close to hand except that the mathematically inclined article has been removed, presumably for correction. Next reference, Saint Nelson (https://www.passlabs.com/technical_article/phase-coherent-crossover-networks/) provides some explanation but is more intended for active crossovers. I am dimly aware that some of the later Tannoys employed a lattice circuit in the high pass section to correct a phase anomaly (by providing a time delay) resulting from the shortened hf horn. (The earlier horns i.e. Monitor Gold used deeper Alnico magnets and so their horn was necessarily longer.) Has anyone on DL had any experience with lattice (as compared to the usual ladder) crossovers?

Another interesting article on crossover designs, this time subtractive crossovers. And more hurt heads I guess.
www.labjc.com/?p=4595

As a taster, the author has harsh words for the Altec Model 17 aluminum horn tweeter, to wit "It is an extreme effort to make these horns barely listenable… but dipped in tar and sand, packed in clay, covered with felt “dots” and bathed in C37 varnish… cut or ground away the dividers, glued “wookie hair” to the edge of the horns… etc. good fucking luck. you’ll need it." Mmmm, where might have I heard similar sentiments before?