Nothing wrong with shooting, throwing, batting, or fencing left handed, but writing with your left hand makes absolutely no sense since you can’t see what you’re writing. Just seems like common sense to me.
I am somewhat left handed. I cannot write at all with my left hand, as I do not seem to have fine control for writing or drawing with it. However, I throw left handed as I have good control and power, but I punch right handed as I do not have control or power to punch with my left hand (different muscle groups I guess). As far as all other mundane tasks (such as eating), I use either hand interchangeably.
Why do you need to see what you are writing? Don't you know what you are writing?
I am right handed but I never look to see what I am writing. I know what I am writing. If you are watching your hand and have to read what you are writing to know what you are writing then you were probably not taught properly.