Batman's pole is on the left... as they access them from his study in Wayne manor. The same pole is on the right as he ascends from the Batcave. This should give you a clue as to how the cave is situated under Wayne Manor. In other words, Bruce and Dick keep moving in the same direction once they reach the cave floor, as they were running toward the sliding panel in Bruce's study.
Also, keep in mind that this arrangement is designed to make the Batmobile readily accessible by putting Batman nearer to the driver's side door and Robin nearer to the passenger side door.