Chris Morriss wrote:

In message , Old Fart at Play
writes
You can keep the low-pass and high-pass outputs in phase
with a 4th-order filter.

Not without an additional all-pass filter you can't. And with an added
all-pass you can make a second order crossover have no phase difference
between the LP and HP sections if you want.

Perhaps you would like to refer to the Loudspeaker Design Cookbook
which has graphs of amplitude and phase for various filters
including the fourth order Linkwitz-Riley filter.

--
Roger.