Speaker phase observation and question?

Vandersteen also uses first order filters which don't honk up phase
...
and his crossovers are incredible feats of engineering without a big baffle to smooth frequency response - a big baffle is a nice flat horn

wiring a driver out of phase to attempt to fix a steep filters phase shift is NOT excellent crossover design...
but the magazine guys love it...

" Vandersteen also uses first order filters which don't honk up phase "

I have 3 pairs. I also had the exact same pair of speakers as the OP, and was able to do a direct comparison. Yes, Vandersteen is a better design, but I don't think that's the problem. If you look at the OP's description of the of complaint, it looks like a problem that needs to be fixed.

" When connected in positive phase (red - red, black - black), the speakers put out pretty substantial bass, but the mids and treble are somewhat subdued. Upon reversing the phase, the mids and treble open up substantially, and the bass becomes somewhat subdued."

I've seen several times where speaker cables were not labeled correctly. If the cables are terminated as a biwire config, and a mistake was made on either the high or low cables, but not both, it sounds like the OP's problem. One way the bass is correct, but the highs are not. After the swap, the highs are good, but the bass is now bad. The problem keeps going back and forth, and if you're not aware of what's going on, It'll drive you crazy.

Keep in mind that this is a guess on my part because we have no idea of what the OP's system is.

Thanks for the informative response Erik.

sfall, I don't believe there is anything horribly wrong with my system, this just confirmed what I was hearing and seeing with the polarity test.

To answer your initial question, my rig is Simaudio pre/DAC, Emotiva XPA 2-channel 300wpc amp, and Audio Art single shotgun run cables, with custom jumpers that mirror the cables.

Although I hear substantial differences between lows and highs when changing polarity, the system sounds amazing either way. I think if I could leave the bass in positive polarity, and reverse the polarity of mid/treble, the sound would be perfect. I should note that I am almost completely deaf in one ear; so when I say a "substantial" change, it my just be that I am hearing frequencies differently than others??

Thank you to everyone else that has taken time to comment, this has been for informative.