General aviation instructor reported a rudder malfunction during climb on a training flight. The instructor took control of the aircraft; returned to the departure airport; and landed safely.

Narrative

Student pilot took off normally from ZZZ off Runway XXL; I was instructing him to level off to avoid the Class B airspace above; level off at 4400' was normal with a shelf above at 5000; once we cleared the shelf I instructed the student to climb to 5400' and level off again to avoid the next shelf at 6000'; at that second level off I felt the plane yaw and I asked what was going on with his rudder; he said he had full right rudder trim in and was pushing the right rudder firmly but it was difficult and felt different. I said 'my controls' and took over flying the airplane. I felt that despite all the right rudder trim I still had to step on the right rudder pedal to maintain directional control. I began a turn back toward the airport and notified the Tower we were having trouble with the rudder and would like to return to the field. They said to proceed to the field and enter a left downwind for XXL; we confirmed. At this point I asked the student to help me out and put some pressure on the right rudder pedal so I wouldn't get tired. Tower then asked us to turn to the south and cross the extended centerline for XXR and switch to a right downwind for XXR; so we confirmed and entered a right downwind for XXR; they cleared us to land and we landed safely; exited the runway and contacted ground as instructed. Ground said to exist and park anywhere. We found a parking spot and shut down the plane. I asked the student to start calling local maintenance facilities and FBO's to see if we can get someone to look at the plane without success. We proceeded to remove the rear panel behind the back seat to see if the cables were clear of obstructions; which they were; inspected the pedals and the rudder trim. We ran the rudder trim through its full range of motion and re centered it. I called my A&P and discussed the issue; I asked if it could be a spring broken and he said he wasn't certain if that model had a spring. We noticed the rudder was centered; pedal travel seemed normal and trim was also centered so we taxied around for a bit and confirmed directional control was normal. We parked the plane and found a mechanic on the airfield and consulted with them; this mechanic mentioned an issue where the trim indicator could get off center. We returned to the airplane and decided to taxi; runup and attempt a takeoff roll. We briefed the takeoff as normal with special emphasis on loss of directional control we would cut power and exit the runway. Takeoff was down with a slower application of power to feel the rudder effectiveness and was normal and we proceeded to rotate and climb out; also normal; cleared the airspace and got on with ATC for flight following. After leveling off at 11500 we noticed far more right rudder trim than normal was required when in level flight but the flight could proceed safely so we opted to return to our home airport at ZZZ1. Upon arrival I attempted to contact ZZZ Tower to see if they had given us priority handling; but Tower does not accept phone calls after XA:30. I feel we adequately handled the situation as it was unfolding; however I also feel I should have taken more time after the issue before deciding to fly back.