It's interesting, after staring at the schematics for a bit, you realize that the structure is surprisingly simple. It just sums a weighted input with the the output of the adder, its first integral and its second integral (pretty obvious in hindsight). The response is:
H(s)=sum( g*vin + a*x + b*x/s + c*x/s^2), where x is the summer output and a,b,c and g are set by the component values and feedback.
This of course simplifies to a plain old biquad, and we can tap at different points in the loop to pick different zeros to get HP (first order zero), BP (second order zeros) or LP response (no zeros). I've look at 3 different structures and they all work the same way.
The insight I'm still lacking is a clear mental picture of advantages and disadvantages of the various structures, particularly when taking non-ideal opamp behavior into account.
For example, FvM suggested the structure here (
https://en.wikipedia.org/wiki/State_variable_filter) as a better implementation. I have not been able to figure out how it's better, except that the tuning equations seem to be more decoupled and easier to handle.