A lot of misinformation and conjecture in this thread.
Reversing the polarity of the tweeter simply changes the phase. What is phase? Phase indicates the relative time arrival of frequencies. Note the word 'relative'. If you had flat phase, it means all frequencies are arriving simultaneously. Crossovers introduce phase warps. Higher the order of the crossover, greater the change in phase.
In a typical two-way, you are looking for phase overlap between the two drivers in the crossover region. Remember that the total output at a frequency near the crossover is the vector sum of the woofer and tweeter output. So, if you have the tweeter output 180 degree out of phase with the woofer, you will get perfect cancellation. If it's in phase, meaning the phase responses of the two drivers overlap, you will get a perfect sum. If it's somewhere in between, you'll get non-perfect addition and subtraction as you move through the crossover region.
Ideally, you want the phase response of the drivers to overlap through the crossover region, maybe until the drivers are down by 24-30 db. This will give you a nice sum. Of course, as you get off-axis, you might notice a dip near the crossover. This means that at that frequency, your drivers are out of phase. This happens because relative to your measurement point, the drivers center to center distance has changed. For example, if you measure up, then the tweeter is closer than the woofer. This will cause the tweeter frequencies to arrive earlier than the woofer frequencies at this measurement point, i.e., a change in phase overlap.
So, when you reverse the polarity of the tweeter, all you are looking for is whether the phase overlap through the crossover region is better. Flipping the polarity changes the phase by 180 degrees at the crossover frequency -- kind of a crude tool, but a tool to improve the crossover nevertheless. Crossovers and other filters on the other hand can change the phase in smaller increments.
Flipping the polarity has nothing to do with compression and rarefaction of waves. If you experience a better "transient" response by flipping the polarity, what you are really experiencing is a better phase overlap, which results in a flatter response (less undulations through the crossover). If you have excellent phase overlap between the drivers, it can sound really good-- like a single driver -- and it will be almost impossible to tell where the crossover is.
EDIT: A point about waves. Normally, we need several cycles of a frequency before we can tell what frequency it is. Basically, we need to hear the frequency for a while before knowing its identity. When a drum is struck or a piano note is struck, it's not like only a single cycle is emitted. Hundreds of cycles are emitted. For example, you can fit 20 cycles of 20 Hz in one second. With some software one could experiment how many cycles at 20 Hz are needed before one can discern that it is 20 Hz (and not 30 Hz or 40 Hz and so on).