In my experience, the saddle is either tilted fore/aft at too sharp an angle or is too high or low.
If the seat is too high, the tendency is to reach for the pedal, at the bottom of the spin. This causes the hip to shift down and the other up, or 'reaching' with the foot.
If the seat is too low, the hip will be driven up at the top of the spin.
Either high or low seat position will create a hip rocking movement if observed from behind.
Assymetric pushing and pulling from right to left can also cause 'tailbone' issues.
I have also observed (primarly with MTB's) people that take a short stroke on one leg, (uphill side) and a long stroke (downhill side) on the other leg, while traversing across a hill.
Also make sure the seat rails aren't bent or broken.
If all else fails PM Finndog for his 'core' workout.