Why do four cascaded all-pass filters cause a 180° phase shift?
I'm here to help since I got the same question some time ago, now I got it. 

What happens is: a first order all pass filter shifts the cut-off frequency to 90º and shifts the f=infinity to 180º. 

If you get 4 filters (with the same value for componentes) cascated, you will get the cut-off frequency to shift not 90º, but 90º x 4 = 360º (at the output). In the end makes no difference in that frequency. 

However, the first filter shifts one specific frequency to 45º (as well as one to 50º, one to 55º ). Four filters in a row will shift that frequency to 45º x 4, which happens to be 180º at the output. 

Now that is the frequency which will have the first notch. 
Other way to get 180º of shift is adding 360º (discovering a total shift of 180º + 360º=540º) because it will have the same phase carachteristics. 
In that way, we do the reverse engineering, and knowing that it will be 540º at the output, we can discover how much does it shifts at each filter (divide by 4 filters, equaling 135º if i'm not mistaken). 

In that way, you can use the equation of the phase of first order all pass filter to discover what frequencies are notched, dependding on the values of your components, substituting the phase on the equation, with 45º and 135º! 

Hope you understood, as I am also doing something along the lines of the MXR-90 for a college work. 

Any doubts you can ask me!