The Girard-Perregaux 350 was the first "modern" quartz movement. It has a 32768 Hz crystal and an integrated circuit to divide it down to one pulse per second, which drives a stepper motor to advance the hands. This is the technology that won and is still used in most quartz watches today.
In addition to other problems (these have notoriously bad calendar mechanisms), this 350 did not always advance the hands properly, even after cleaning. It would sometimes step backwards or just twitch.
The motor turns a half turn each impulse, and depends on the permanent magnet to orient it correctly between impulses. If it does not turn perfectly freely, this does not work reliably and the next impulse may not turn it the right direction, or at all.
Upgrading the worn and sticky old teflon bushings in the stepper motor to correctly sized jewels fixed it.
Another watch, with a 354, that came in at the same time had harder teflon bushings, and it worked with no trouble after cleaning. The 354 also has a reengineered calendar mechanism.