B737 Captain reported excessive trim was required to maintain stable flight and decided to divert for landing.

Narrative

After rotation the plane felt 'squirrely' but I figured it was just very windy. As the aileron deflection required was excessive I was evaluating whether the wind was causing the flight control manipulation required or was the trim off. I scanned the trim settings and everything was centered. As we climbed I determined that lots of rudder trim was needed. I hand flew the airplane whilst attempting to bring it into trim. As the amount of trim needed was getting excessive we scanned the engine indications and flap configuration to see if we had an issue and everything was normal. We continued the climb as we were evaluating the issue. The fuel was balanced and nothing appeared abnormal; we had no hydraulic; feel control warnings etc. Once I reached about 5 units of Left rudder trim the AC was able to fly more level but still had about 2 units of Left aileron deflection. Eventually; around 7 units were needed for a relatively level flight. I engaged the auto pilot and discussed the situation with the First Officer (FO). I passed Pilot Flying (PF) duties to the FO and had him see if he concurred with the present control configuration (the trim we had was required) and he did. I contacted [maintenance] via ACARS and [flight operations] informing them of our situation. Now at altitude I decided to disconnect the Auto Pilot and while hand flying had the FO slowly take out some rudder trim to evaluate if I had mistrimmed or over trimmed. It became apparent quickly that the trim was needed as aileron deflection increased correspondingly. We discussed that the trim was now about where it was when we pre flighted the aircraft. We now thought that that was where it was left by the previous crew and not inadvertently moved as originally thought. A quick review of the flight manual revealed no non normal procedure for this situation. We discussed the situation and whether or not to divert. As this was way out of the normal range for trim and no other abnormalities were present we suspected that the aircraft had a flight control rigging issue. We were concerned that if this condition worsened or we developed any other issue we could easily be put beyond the ability to control the aircraft and decided to [request priority handling]. I had no response from [maintenance] at this point. We made the decision to [request priority handling] and divert. I choose ZZZ1 as it was close; had a long runway; emergency services and lastly customer services. We knew this was going to require an overweight landing but did not think it prudent to burn down as that would have taken hours. Dispatch was informed and quickly got back to us acknowledging the divert and situation. We briefed the plan; the approach; got landing data and considerations for the overweight condition. We conducted an overweight ILS Runway XXR and had a smooth uneventful landing. Emergency responders confirmed that our brake temperatures were good and we proceeded to the gate.