PA-28 Flight Instructor practicing crosswind takeoffs with a student reported the student lost control of the aircraft resulting in a runway excursion prior to lifting off.

Narrative

While working on crosswind takeoffs with private student pilot; at roughly rotation speed; the student was well left of centerline on runway with left crosswind and changed from left aileron deflection to right aileron deflection. Instructor vocalized to remove right aileron and student removed all right rudder input instead; resulting in instructor taking controls. Instructor was committed to rotating at that speed and put in one notch of flaps to help rotate but did cross the runway edge lines before rotating. After becoming airborne and stabilizing back on runway centerline; instructor maintained controls and returned to land to inspect runway edge lights and inspect the aircraft. No sounds or impacts were heard/felt on rotation but just wanted to confirm. No markings or damage to grass/runway edge lights near the runway noted and no damage to aircraft gears; propeller or underbelly noted after taxi and shutdown. Instructor debriefed the student about what happened; the importance of maintaining centerline on takeoffs and landings by using right rudder to counteract left turning tendencies on takeoff and also discussed the miscommunication between aileron deflection and rudder deflection. Removal of all right rudder while already being left of centerline allowed for aircraft to go past runway edge markings before fulling rotating off the ground. Instructor also let the student know that the fault was with the instructor; not the student; and that the instructor; while having their feet on the rudders as well; failed to properly guard the right rudder during rotation and allowed the student to drift left of centerline; trying to give time/opportunity for the student to correct the situation. In the future; student will strive to maintain centerline better and not accept drifting off-center; and instructor will better guard the rudder on takeoff and not allow students to drift off centerline - will require immediate corrections in future from students.