There's a certain author, I can't remember who, that they say learned to be ambidexterous because his cat would fall asleep on his dominant arm while he was writing, and he didn't want to wake her up!
I certainly think it's possible, just frustrating. One thing I learned when I started doing henna is that I can do a perfectly good henna design left-handed, it just takes me twice as long and is twice as tiring.
Here's a photo of a design I did on both my right and left hands. I always wondered if it would be the same with handwriting vs. a repetitive, decorative design, since the alphabet is a little stricter than flowers and paisleys.