B737-700 Captain reported the control wheel got stuck in a 45-degree roll position and the stabilizer trim malfunctioned.

Narrative

We pushed from Gate XX in ZZZ; started both engines; and performed our flows and Before-Taxi Checklist. VNAV and LNAV were already selected by the FO (First Officer); having programmed the FMC using the 2L/3L method in the preflight planning. The MCP (Mode Control Panel) was the older type that we don't see very often. We began our taxi onto [Taxiway] XX as instructed by Ground Control; to depart Runway XXC. Abeam Gate XY on Taxiway XX. The control wheels rolled left 45 degrees and the stab trim began trimming nose down to approximately 3.8 units - top of the green indicator band - then stopped and reversed; the stab trim then trimming nose up. I retarded the thrust levers and began braking to stop the aircraft. I reached over to the control wheel with my right hand and it felt stuck in the 45-degree roll position. At no time did the control column move forward or aft. I asked the FO to do the same and he agreed it felt stuck. He also looked at the overhead and forward panels and stated no lights or indications were illuminated or inaccurate.The stab trim was still trimming nose up; approaching the bottom of the green indicator band. I reached over to trim nose down with the control column trim switches; but this did not arrest the trim. The stab trim continued trimming nose up out of the green band to approximately 9.0 units; where I arrested it by selecting both stab trim cutout switches to CUT OUT. The autopilot disengage red light flashed; accompanied by the aural warning tone. We also noted the Stab Out of Trim light illuminated. The control wheels centered.We used the risk management model to work through what had occurred. Why/how was the autopilot engaged? Why did the trim only travel to the top of the green band; 3.8 units; before reversing; similar appearance to a test sequence? We agreed that returning to the gate was the best course of action. I notified the passengers and crew of such and the FO contacted Ground Control for our clearance. We returned to Gate XX where we notified ZZZ Maintenance. I contacted my Dispatcher.