Relative to the handedness, one bolt is always moving with it, one bolt is always moving against it. Switching direction doesn't change that. So switching direction can't change whether it moves inward or outward.
Couldn't you transform twiddling one way into twiddling the other way by reflecting through a mirror and turning the bolts 180 degrees end-over-end? If you start with the bolts moving North, the mirroring makes them go South, and turning them around makes them go North again, but now you're twiddling the other way. So the bolts would have to go North no matter which way you twiddle them, so either twiddling is not reversible (which is a repugnant proposition) or they must have a speed of 0.