<font color='#000000'>1st order xovers are great if (and it is unfortunately a big "if"
the drivers can handle the power outside their frequency range.
Most speakers will be (relatively?) fine at low levels, since there is no stress, but as power increases, so does the demand on the drivers. The two main problems are tweeter overload and mid-woofer cone breakup - both of these are minimised with higher orders.
A 12dB xover
does reverse the phase of one driver, but this is fairly consistent over a wide frequency range, so the reversed driver is actually in phase with the other. A 24dB L-R xover reverses the phase twice (i.e. a 360 degree shift).
Note: This is
not the same as "no shift" - there is an effective delay of one cycle.
Odd order filters (6dB, 18dB, etc) have a relatively constant 90 degree shift between the two outputs at all frequencies, with the HP output 90 degrees in front of the LP output - this is only relevant for steady state signals though
Correction: An 18dB filter has a 180 degree shift caused by the 12dB section, and a 90 degree shift from the 6dB section, so the total between the HF with respect to the LF signal is to effectively create a 270 degree shift, and the output of the HP section is now
lagging (i.e. 90 degrees after the LP section). Again, this only applies to the steady state signal.
Much has been said about transient response, but in an AB test of a summed 24dB crossover and the original signal, it is virtually impossible to pick which is which - it shows on an oscilloscope with a square wave input, but still sounds the same.
There is still much work to be done in this area - I certainly don't have anywhere near as many of the answers as I would like, and so far, every new answer has raised new questions
</font>