If the door can be reversed (and that's a big if) then I can't imagine it can be done without removing the entire door frame. The fixed pane is, well, fixed and unless you see a way to detach it from the frame and attach it to the other side then you may be out of luck.
So let's assume you can't remove the fixed frame. Well, you could try rotating the door. This should work, but it would put the rail on the outside of the door. I don't know how this exposure would affect the longevity of the product, and you'd certainly be prone to more things getting built up in the track. This also means that the exterior side of the glass would be facing the interior. A lot of modern glass has reflective coatings and treatments on the outside for energy efficiency purposes. You'd be losing this, and likely by making it worse in the summer (as energy would more readily penetrate and less readily escape).
Lastly, if your door is flashed and sealed, and if it has a flange around the frame, it might not even fit the other way without sticking out of the siding, and if it's vinyl it likely has drain holes in the frame which would now drain into the house.
You'd then have to reverse the lockset and handle on the door. This is probably the easiest part.
But, in conclusion, you should recognize that you would likely:
- Need to remove the entire frame from the opening
- Likely lose energy efficient properties from the glass treatment being reversed
- Expose your rail track to the exterior and additional wear
- Need to reverse your handle and lockset (and lose the ability to use the "bar in the track security method)
- Need to re-flash and re-seal the door (provided that there is a flange and that it's still usable)
In my humble opinion, each of these together would warrant simply buying a new sliding door to replace this one. Because the door frame isn't really all that expensive. The issue is the labor of removing, replacing and resealing the door.... which I think you'd have to do either way.