ACC_Expat, where were you when I was a kid? Life would have been much easier if I knew I wasn’t the only weirdo. 
yanceylebeef, I know exactly what you’re talking about. The word is not ambidextrous (doing stuff equally well with both hands) but cross-dominance (doing certain activites well for different hands).
I am cross dominant (Luke Skywalker voice: as my father before me). I write with my right hand but I swim as a southpaw. My swim instructor gave me a lot of grief because I couldn’t do the scissors-kick to tread water. When I reversed the motion of my legs, voila! – treading water ‘left-handed’ became easy.
I read this book a long time ago – can’t remember the title or author – but the point was that you can exercise your brain by doing things with your non-dominant hand, which will also develop both sides your brain equally (so the theory goes). You can take this beyond hands: if you listen to the phone with your left ear (meaning the creative right side of the brain), switch it to you right ear (meaning the more logical, left side of the brain).
Nowadays, I can almost do certain tasks with either hand, such as 10-keying, using the mouse, cutting cards, shooting, and eating with chopsticks or a fork (I hate the stupid knife-fork-hand-switch when eating steak, but that’s a rant for another day).