SR22 flight instructor on training flight with student reported aircraft loss of control and runway excursion during takeoff roll.

Narrative

As my student pilot was preparing to takeoff and applied full takeoff power; the airplane began to veer to left from the left turning forces. I began to give verbal direction to input right rudder to counter the left turning forces. However; the aircraft continued to veer off of centerline and I attempted to apply right rudder but the airplane did not correct course. At this point; I took over the airplane controls and pulled the throttle back to idle but could not redirect the airplane in a timely manner to remain on the runway. As a result; we rolled off the runway and into the grass and then taxied back onto the runway. Immediately following; we taxied over to parking to discuss what and why the loss of control occurred. It turned out that cause of the loss of control was due to the student pressing on the left (i.e. wrong) rudder pedal instead of the right rudder pedal. This action effectively acted like a jammed control and prevented me from correcting the veer to the left. A contributing factor to rolling off the runway was my delayed reaction to cut the power. I estimate that if my reaction time was a few seconds earlier; I likely could have prevented the airplane from rolling off the runway. Lesson learned for me is to guard the power lever more closely when I have a low time pilot and assume that the student pilot can get confused on which input to apply on the controls. I will also train with the student to practice rejected takeoffs if something is wrong during the takeoff roll.