PA-28 Flight Instructor reported a loss of control during landing as the Student Pilot relinquished the flight controls after encountering a strong crosswind gust and pushed the aircraft off centerline. The reporter performed a go-around and diverted to another airport where a ground observer could assess the status of the landing gear and subsequently performed a normal landing. Upon landing; the aircraft was found to be damaged in several places.

Narrative

During an instructional flight the student relinquished control to me on landing on Runway XX as the crosswind; 190 at 5; gusted and pushed the aircraft to the left of centerline towards the left side of the runway. After I took control of the aircraft in its dirty configuration; the speed was at a stalled point. The aircraft came down and bounced left of centerline and the right wing was in a right banked attitude. I did not immediately pitch the nose down as it would have caused the propeller to impact the runway. We proceeded across the runway to the right side of centerline and bounced in the grass. As the aircraft became airborne again; I lowered the nose to regain just enough airspeed to regain lift and keep the aircraft airborne and climbing. I regained centerline at full power; turned off the carburetor heat and began retracing the aircraft flaps 10 degrees at a time to a normal climb. Not fully knowing the state of the landing gear of the aircraft; I staved off landing at ZZZ and returned for a landing at ZZZ1 whereas someone could assess the landing gear after a precautionary slow pass. After a precautionary landing of the aircraft; damage was discovered on the leading edge of the left side of the stabilator; the bottom of the right stabilator; and the underside and trailing edge of the right wing tip.