I have played that way (right-handed stringing, played upside down) since I was given a guitar at the age of 10. Being left-handed, I just picked it up and played it, not realising that the stringing was "wrong" for me.
When my best friend (right-handed) got a guitar a few months later, I could play his and he could play mine, which was very useful. The chord diagrams look exactly right for the way I play.
I played in bands using a electrified acoustic guitar. When I bought my first solid electric (an Antoria Gibson SG copy) it was left-handed, so I thought I'd try to learn to play "properly" left-handed. This meant throwing away over 10 years of playing experience so I abandoned the idea and converted the SG copy to right-handed stringing. This meant fitting a new nut and reversing the angle of the bridge. I used that guitar for many years.
Nowadays, I have a left-handed Strat which was easy for a local dealer to convert because the bridge has enough adjustment in it. It just needed a new nut and a new set of strings. I also recently bought a genuine SG Standard, again left-handed. I ordered a new scratch plate for it with the post holes for the bridge reversed - I drew round the existing one then turned it upside down to get the new position for the bridge posts then send it to a company that makes specialised scratch plates. It was fairly cheap and a quick turnaround. I then gave the guitar and the scratchplate to the same dealer who converted it.
I have not removed the scratchplate to see how it looks underneath but the dealer said that he's fill the old bridge post holes with mahogany dowling or similar, then re-drill them. Anyway, it plays beautifully.
Any questions, feel free to contact me on [email protected].