When Neil leaps off the ground, and then he starts to swing his arm, the rock in the stocking resists the aceleration he tries to make it undergo. As his arm pushes on the stocking/rock, the stocking/rock pushes back against his arm, and he is turned in the opposite direction of the swinging rock. He feels what we call torque. His turn rate is exactly proportional to the rock's, each would find that the rotational inertia times the spin rate was the same, (smaller rotational inertia, larger spin rate). Because the moon's atmosphere is almost a perfect vacuum, once the rock is up to speed, it needs no further push, so the Neil-rock system will simply spin until Neil falls back to the moon's surface, where his feet will grab the surface, he will stop spinning and the rock will start to wrap around him and eventually whack him.
This can be done on an icy pond on earth. You will probably, scientifically, fall on your butt!