Totem Hawk crossover design. Why tweeter polarity is reversed?

Capacitor shifts Tweeter phase 90 degrees one way.

for frequencies approaching 0,

Inductor shifts Woofer phase 90 degrees the other way.

for frequencies approaching infinity.

If the cut-off frequencies are the same, the phase difference between a first-order high-pass and a first-order low-pass is 90 degrees at all frequencies. That doesn't take into account things like driver response and differences in the distances of the acoustic centres of the drivers to the ears of the listener, though.
 
for frequencies approaching 0,



for frequencies approaching infinity.

If the cut-off frequencies are the same, the phase difference between a first-order high-pass and a first-order low-pass is 90 degrees at all frequencies. That doesn't take into account things like driver response and differences in the distances of the acoustic centres of the drivers to the ears of the listener, though.
Yes, so?
At crossover frequency shift is significant
Inverting tweeter phase when using these simple crossovers has been done for decades.
But if you think it is not worth it, don't.

In any case, my point is showing the OP that "there is an inductor hidden in plain view" which he didn't consider.