The pilot reported that while conducting touch-and-go landings in the traffic pattern, the gyrocopter touched down on the main landing gear. When the nose wheel contacted the runway, the gyrocopter swerved abruptly and began bouncing left-to-right, then rolled over, resulting in substantial damage to the empennage and main rotor system. The pilot could not recall the airspeed when the nose wheel made contact. He further stated that the nose wheel may not have been aligned with the direction of travel at touchdown, which could result in an abrupt loss of control. The pilot added that the nose wheel steering was connected to the pedals and rudder system and that both systems were functioning normally prior to the accident. The pilot reported no pre-accident mechanical failures or malfunctions with the gyrocopter that would have precluded normal operation.