Linkwitz-Riley Crossover Sum as Allpass Filter
Just adding to the answer. The sign for recombining L/R filter alternates, so it's '-' for second order '+' for 4th order and so forth.
The algebra is isn't all that pleasant so I will only go through the 2nd order. A second order L/R lowpass is simply the cascade of two first order Butterworth lowpass. The first order butterworths are
$$L = \frac{1}{1+jx}, H = \frac{jx}{1+jx}$$
where x is the normalized cutoff frequency $x = \omega/\omega_0$
That makes the difference of the L/R filters
$$H = L^2-H^2 = \frac{1^2}{(1+jx)^2} - \frac{(jx)^2}{(1+jx)^2} = \frac{1+x^2}{(1+jx)^2}$$
The numerator is real, so we just have to proof that the magnitude of the denominator is the same.
$$|(1+jx)^2| = |(1+jx)|^2 = (1+jx) \cdot (1-jx) = 1 + jx =ix + x^2 = 1+x^2$$
which indeed equals the numerator.
Another interesting class of &quot;recombining to allpass&quot; are odd order Butterworth filters. Here the lowpass and the highpass are 90 degrees out of phase so both the sum AND the difference of the filters will be create an allpass, although one has significantly more group delay than the other and the &quot;optimum&quot; choice of the sign also alternates with order.
Odd order Butterworth also have only -3dB of gain at the crossover whereas L/R have -6dB.