Some shifter dogs can be reversed. It depends on the design. If both sides are exactly (EXACTLY) the same, most likely. If there are slight groove marks on one end, it can not.
Many shifter dogs are designed whereas they could be installed in either direction BUT MUST NOT be. If not installed properly, a new shifter dog would cause the unit to jump out of gear on the first run (this does not apply to the 1960 model).
Machine shops have been known to restore (cut away rounded area) the gear and dog lobes to their straight sharp edges.
YOur forward gear is part number 377174 but has been superseded up to part number 381810.
Your shifter dog is part number 305105 and is not superseded. However you can also use the later model shifter dog which is part number 312866.
The 312866 dog incorporates the newer indent spring and balls to help keep the dog in whatever position one shifts it into BUT it can be used simply by NOT USING the spring and balls.