Well, think a bit more about the direction of the torque applied to the nut by the pulley.
For CW spindle rotation, the pulley is tending to slip in the CW direction, which will tend to tighten a RH threaded nut.
The only cutter decks I've encountered LH threads on, have CCW spindle rotation. The old Mobilco Heavycut slashers spring to mind, and most Greenfield ride-ons.