B737 Captain reported aircraft controllability issues after takeoff. The flight crew elected to request priority handling and divert to make a precautionary landing.

Narrative

After takeoff and cleanup from ZZZ airport Runway XXL; First Officer noticed aircraft rolling left. While climbing began to use rudder trim to correct situation. Rudder trim began to yaw the aircraft but the aircraft was still rolling left. We continued to climb; Above 10;000 ft. we used aileron trim to stop the roll. We attempted to trim out the roll but as we climbed and accelerated the roll left continued. We decided to level at FL310 since the problem was not improving. We were able to trim the roll out; but the yoke was deflected 6-8 degrees right to hold level flight. Contacted Dispatch and phone patched to Maintenance and Chief Pilot. All decided diverting to ZZZ1 for overweight landing was best idea. We were approximately 150 miles south of ZZZ1 at FL310. Used diversion guide and referenced overweight landing. planned a long final ILS to XYL. I flew the approach and landing and First Officer monitored. Taxied to Gate. First Officer did an excellent job. Good flying skills and procedural knowledge.