The flight instructor and his student were practicing hovering autorotations over an asphalt taxiway. The flight instructor reported that during the third autorotation, he either did not completely roll the throttle into the spring-loaded override, or he inadvertently allowed the throttle to open while arresting the descent. The helicopter climbed to about 25 ft above ground level and then it entered an uncontrolled right spin. The flight instructor immediately communicated to the student pilot that he had taken over the flight controls but was unsure if the student pilot had fully relinquished the flight controls. The flight instructor maneuvered the helicopter over a grass field and initiated a hovering autorotation. The helicopter landed hard and spread the skids, which resulted in substantial damage to the fuselage. The flight instructor shutdown the helicopter and he and his student were able to egress without further incident. The flight instructor reported that there were no preaccident mechanical malfunctions or failures with the helicopter that would have precluded normal operation.