R. Douglas Fields in Quanta:
Dazzling intricacies of brain structure are revealed every day, but one of the most obvious aspects of brain wiring eludes neuroscientists. The nervous system is cross-wired, so that the left side of the brain controls the right half of the body and vice versa. Every doctor relies upon this fact in performing neurological exams, but when I asked my doctor last week why this should be, all I got was a shrug. So I asked Catherine Carr, a neuroscientist at the University of Maryland, College Park. “No good answer,” she replied. I was surprised — such a fundamental aspect of how our brain and body are wired together, and no one knew why?
Nothing that we know of stops the right side of the brain from connecting with the right side of the body. That wiring scheme would seem much simpler and less prone to errors.
More here.