In my 67 I have a 4 speed manual transmission, in every picture that I see of a 67 the reverse gear is toward the rear of the car...but my reverse gear shifts toward the front of the car? What 4 speed is this?
The type of shifter can make a difference too. In my 68 with a toploader the stock shifter had reverse to the back. When I put a Hurst shifter on it reverse was to the front. The Hurst flipped the reverse lever on the transmission. I assume this was done to avoid accidentally shifting from 1st to reverse under hard acceleration.
THe 64 1/2 I restored a few years ago had a toploader with "reverse up" or forward. By the 65 model year this had changed to the traditional reverse down position. Sunbeam Tigers also had the reverse up shift pattern. I see two possible scenarios that could explain your situation if you have a toploader transmission.
1) Less likely scenario. I assembled my 64 1/2 from pieces including a shifter aquired on ebay. It took lots of time to get it working as the flat steel arms (on the transmission) had to be oriented properly pointing either up or down and the shifter and rods had to all hook up and work without interferance AND fit into the car. I seem to remember playing with the setup and it may be possible to flip the flat steel arm mounted on the transmission for reverse from its current up orientation to a down orientation (or vice versa, can't remember how they are set up). This change would change the throw of the shifter from forward/up to backward/down. It could be someone took your car apart to rebuild the shifter and they accidentally reversed the orientation from stock, then adjusted it to work. The clearances were tight so this may not work, but is an idea to consider.
2) More likely scenario. Maybe someone swapped in an earlier transmission shifter setup?
I do recall my Hurst shifter on my '69 being reverse up as well.