To answer your question, my AGWAY is built by Murray. They all have progressive pulleys (if they are variable-speed), with a foward/neutral/reverse transaxle. (mine also has a hi/lo gear in it).
You probably have a system kind of like the JD RX75 RER, where the selector adjusts the pulley in foward gears, then pulls an arm (directly linked to the transaxle) to go into neutral and reverse. No, you can't just change the selector, because it's not a disk drive - it's a gearbox that gives you fwd/rev. motion.