I'm the same, just mirrored.. I'm technically right-handed, but my left hand works as well as the right one for stuff that I haven't exclusively trained my right hand to do (or the left one, for that matter, as far as the right hand is concerned). So I can't write with my left hand (except that I can mirror-write, especially if I write with my right hand at the same time. Apparently Leonardo Da Vinci used his left hand when he mirror-wrote, and his right hand when he didn't. Read that somewhere).
I'm also like that. When I was a child (not sure what age), I would take two pencils and write one line with the left hand, and the next with the right, indistinctly. But my school teachers said I should choose one and stick to it (not sure if that would still be advised today). I really had no preference so my mother advised me to choose the right, as it would give me less trouble due to everything being designed for right-handers, etc. A reasonable advice.
I did so, and now I wouldn't be able to write with my left hand, but when I take up a new activity, it's still basically arbitrary. For example, I started playing golf recently, and I deliberately chose the right-handed way for the same reasons as for writing (it's easier to find right-handed clubs, etc.) but I could have chosen the opposite and I suppose my skill would be the same, initially there is no option that looks more "natural" than the other for my body. And with trivial things where I didn't choose consciously, sometimes I'm told I do them as a left-hander, e.g. I stir tea counterclockwise.
I'm a fountain pen aficionado and I would have liked my son to enjoy my fountain pens someday, but he's a clear left-hander, so he probably won't :)
