Interesting you have no responses from 9:07 this morning! You will do much better in the gunsmithing forum, with instructions and pictures.
For now, to answer your question, the part that rotates the cylinder is called a hand. You can almost always tell that the hand spring is not working properly if the cylinder will not turn when the gun is pointed up, but the cylinder turns properly with the muzzle toward the center of the earth. There are two studs on the hand, one that fits into the trigger and one that the handspring presses against. My guess is that either the handspring is broken or the stud is broken. Both are very durable and this seldom occurs. Changing hands is common and an easy fix. You need to remove and measure the width of the tip of the hand that turns the cylinder. It should be between .092 and .099, hopefully more toward the .092. If your timing is still good, replace with the same size hand, if that needs fixing too, get a slightly wider one. I'm sure that there are utube videos that will help you with this. If the hand studs are still good it is the handspring and that is harder to replace. Still, it is an easy repair for those with experience. You might want to seek out a gunsmith or armorer if you are not familiar with the insides.