According to the pilot, he was practicing touch-and-go takeoffs and landings as part of his flight training curriculum. The pilot reported that the two previous takeoffs and landings were performed with no issues; he also noted an existing crosswind during the takeoffs and landings. He took off again, and noted that when he turned downwind the airplane was a little "fast." He was able to slow the airplane down to 75 knots, but felt that he was a little high, so he applied full flap and slowed to 70 knots. He reported that there was a slight left crosswind and he corrected with left aileron and right rudder to maintain the approach on runway centerline. The airplane touched down to the right of the runway centerline, he applied left rudder to return to centerline and "verbally confirmed his checks," 'aircraft under control, flaps zero,' and reduced the flap setting to zero. The pilot applied full power and as he was about to check the instrument panel the airplane pulled to the left, which he felt was abnormal. He reduced the power in order to regain aircraft control back to centerline. There were no other anomalies encountered, and the pilot indicated he had enough remaining runway to safely takeoff. He rechecked the instruments, applied full power, and right rudder. About 3 seconds later, the airplane pulled "violently" to the left. The right wing dropped and the airplane rotated to the left. He reduced the throttle to idle and applied full right rudder to no effect. As the airplane was departing the runway surface, he applied brakes. A Federal Aviation Administration inspector examined the airplane, and reported that there were no mechanical anomalies that could have caused the loss of control. The total distance traveled from the onset of skid marks on the runway to where the airplane stopped on a taxiway was about 800 feet. The pilot reported to the FAA inspector that he landed hard to the left of the runway centerline. As he powered up, the airplane pulled to the left approximately 45 degrees. Skid marks from the nose wheel and right main tires were evident for about 300 feet arching to the left. He applied full power, and the airplane pulled to the left again. The left wing rose as the airplane turned abruptly left on the runway. The airplane continued off the runway where it hit a runway marker sign with the right wing. Witnesses reported that the airplane disappeared in a cloud of dust and traveled another 485 feet leaving only one tire mark before it struck a drainage ditch and came to a stop on a paved taxiway. The right main landing gear separated and was located about 200 feet from the point where the airplane departed the runway surface.