It seems that there is no way to set a due date for a task in Reminders on Mountain Lion. In Lion in iCal you could set a Due Date that was different than the reminders you could set for a task. It seems obvious to me that reminders are different than due dates. Reminders can occur multiple times leading up to and on the due date (and perhaps even after a due date), but it is helpful to set a due date. Hopefully Apple will bring this functionality over to Reminders from the Lion version of iCal, as this is a downgrade in this one respect for me.
This seemed to work perfectly for me. But I can't make it happen again.