It's possible that you're more ambidextrous than you realize. I think most people with a fully dominant side would have experiences like Slartibartfast described.
I can generally switch back and forth with anything that requires gross motor skills with relative ease. I was a left handed child but my early teachers forced me to write right handed. My handwriting is pathetic, btw. I have one child who is fully ambidextrous (he switches his writing hand when he gets tired, and it's impossible to tell what he wrote with what hand), one who's left handed, and one who's right handed. It really is interesting to watch them tackle new tasks.
That is really cool. Part of my job is taking minutes, and I would so love to be able to switch hands during the day! Oh, the envy!
I know! I'm more than a little jealous of him in that regard. It was sort of a process though. When he was in Kindergarten he would work the left side of a page with his left hand and the right side with his right hand. In 1st grade (the first time, he repeated 1st), his teacher made him choose a hand. Since every other kid in the class was right handed, of course, he went with his right. He still did everything else with both hands, but kept using his right for writing for a couple years. Then two years ago he developed a large bone tumor that gradually locked up his right wrist. He began using his left hand again. The doctors told me they'd never be able to restore full range of motion in his right hand, so I encouraged the left-handedness, since he doesn't have any deformities in that hand/wrist. After about 6 months of trying again, his left hand writing is indecipherable from his right hand writing.
On a side note, he goes into surgery on Monday for the first of several procedures to attempt to remove the tumor and straighten out the radius (which is curved as a result of the tumor growing and pushing it outward). We've moved to a different hospital, and this hand surgeon wants to use a different technique than his old hand surgeon, and she thinks we might be able to restore 80% of his range-of-motion, which double what the old surgeon was aiming for.