Two ideas.
1) This is a classical application for a Predetermining Counter. Google "Predetermining Counter" and you will come up with dozens of manufacturers. This is the bulletproof approach used in industry for this type of application. You set the dials for whatever count you want. Press reset to prepare for counting. When the count reaches zero you get a contact closure or opening that controls a power relay. Many coil voltages are available to actuate each count, but line voltage would be simplest. You have to mechanically rig up a microswitch to produce a pulse for eash rotation.
2) Since you don't have to deal with a lot of differing situations and things are happening pretty slowly, a cheap and dirty method of stopping the machine at 10 counts would be to have a piece of very flexible, small diameter cable or nylon cord that winds up on the workpiece as it turns and activates a switch when the proper length of cord has been wound up on the workpiece. Perhaps not very elegant or precise, but probably more than adequate to get your desired number of twists. The cord could pull two contacts apart, pull out an insulating block holding two contacts together, pull out a conductive block maintaining conduction between two contacts, pull on the lever of an industrial limit switch, etc., etc. You would just have to be sure that any exposed voltages on contacts were safe and that there was sufficient overshoot available on the switch that damage would not be done as the machine coasted down. But, I imagine that your twisting machine doesn't do much coasting after power is cut. Your contacts would control a power relay controlling the motor.
awright