The paper "Circuit Theory" shows and describes some figures with the two-way crossover's work. At frequencies way above the pass band, the rate of increase in attenuation is 12dB/octave at a 90 degrees phase shift which goes to 180 degrees at very high frequencies. The rate of the attenuation here depends on the filter order, which in turn is determined by the number of reactive components in the ladder. For instance in our case here there are two reactive components, hence the filter order is two and its rate of attenuation is given by nx6dB/octave = 12dB/octave since n=2. Its response at different values of n is as shown below.
A two-way crossover has a combination of a high pass and a low pass filter which could be used to drive a tweeter and a loudspeaker at the same time. These two could be fed from the same model of an amplifier if and only if it can accommodate the frequency ranges of both of them. However this is a wide range hence it’s impossible to come up with such an amplifier, hence the two-way crossover is used.
With the low pass filter designed as explained earlier its high pass counterpart can be derived from it following the simple fact that their frequency response is reciprocal of one another. This means that attenuation of a low pass filter at say a frequency of w=2 is the same as the equivalent high pass at w=0.5. Deriving from this the high pass filter components are the reciprocal of the normalised low-pass filter, such that where there are capacitors in the low pass model they are replaced by inductors in the high pass model.