C172 Flight Instructor reported engine roughness and directional gyro malfunction resulted in diverting to a safe landing.

Narrative

On the morning of Date a private pilot and myself conducted an instrument training flight departing from ZZZ. We filed an instrument flight plan to ZZZ1 and a subsequent flight plan to ZZZ2 since reported and observed conditions provided great actual IMC training for the student. We performed a thorough pre-flight inspection in accordance with the Cessna 172R pilots operating handbook and found no discrepancies with the aircraft. Following our pre-flight we briefed our training flight; which was planned to depart ZZZ; fly the VOR-A into ZZZ1; pick up another clearance to then fly the ILS XX into ZZZ2; and fly under VFR back to ZZZ; weather permitting. Run-up and pre-takeoff checklist items were performed and the aircraft operated normally and was properly configured for departure. The take off and climb up to cruise altitude was uneventful. Once reaching ZZZZZ (IAF/FAF for the ZZZ1 VOR-A approach) we elected to hold as published for training purposes as we were IMC. At this time we experienced two 2-3 second high pitch engine vibrations within a span of approximately 2 minutes. These vibration events were unprovoked and remedied itself. We observed the aircraft's engine gauges and attempted to replicate the issue. We could not replicate the issue and found the aircraft to be operating normally; so we decided to proceed with the flight. ZZZ Approach allowed us to change to the CTAF frequency once we were established inbound. After conducting the approach into ZZZ1 and performing a touch and go we climbed into IMC conditions around 2;400 ft MSL. Shortly after entering IMC we observed the directional gyro precess at a rate of approximately 30 degrees within 5 minutes. While we still had an operational magnetic compass we requested to deviate to our home airport ZZZ and end the flight. During our level off at our cruise altitude we began to experience engine vibrations and roughness. The vibrations were stronger and more prolonged than the previous ones we experienced during our flight. The engine vibrations occurred approximately 4-6 times lasting 2-4 seconds within a span of ~5 minutes. At this time we could not rely on smooth engine performance. We requested vectors to ZZZ3 and the lowest altitude in hopes to proceed in VMC. We were able to establish visual contact with the airport and were cleared the visual approach. We flew a higher and closer pattern than normal in case normal engine performance was lost. The landing was uneventful. Following the flight I made contact with the flight schools chief mechanic and was advised to perform an engine run-up. The engine ran rough during the run-up however the magneto drop was within normal perimeters. I passed this along to the mechanic and we suspected a stuck valve. A local mechanic shop performed an engine run-up and compression test but found no anomalies. 'Av blend' oil additive was also put into the engine. The aircraft was deemed safe following their inspection and a logbook endorsement in the aircraft logbooks was entered permitting it to return to service. The reposition flight from ZZZ3 to ZZZ was uneventful and the aircraft performed normally.