sigh... I just looked at the signals again with the logic analyzer and it is using a form of PWM. I must not have been triggering correctly before or the pulses were too long for my sampling rate.
Longer pulses at the beginning of movement and shorter pulses at the end of movement. The minimum pulse width is 1ms, and it looks like a full period is 8ms (125Hz). It can be on for the whole period or just 1 ms. Does this mean 8 PWM values?
Now that I have this data, what else should I learn from it to use in my own control? I will not be using hardware PWM because I don't have enough (I would need 20) so I will be emulating it in software. So understanding how it is currently working will likely help me with that.