The left side of the brain controls the right side of the body. It also controls language. There's a theory that that's why the words for the left hand (and left in general) are usually associated with evil ("sinister" in Latin means "left", for example), while words for the right are associated with good ("dextrous" comes from the Latin "dexter", meaning "right side"). The left side of the brain might be literally disparaging the side of the body it doesn't control, while praising the one it does.
Yes and no: different parts of the brain DO have different functionality and the language center IS in the left half, but you'd need your whole brain to demonize body parts since a lot more than just language would be involved in the reasoning, the decision, the planning and the execution..
In general the left brain does language. It's been proven by observing people who had their hemispheres separated. The whole story about the left brain calling the left side of the body names is pretty speculative though.
It's also true that we tend to exaggerate the whole left/right brain thing in popular culture. The idea of left or right-brained personalities is one example of a pretty inaccurate characterization of how the brain works. But there are very real differences between the hemispheres.
All of that is just based mostly on recent interviews with brain scientists I've listened to, I'm not an expert!