In scientific literature, the left hemisphere is termed often "the left male brain" or, "extreme male brain" (see Simon Baron-Cohen) to describe man's concern with hard system as opposed to women's concern with more intuitive stuff. There are small pieces of support for this, gender wise, such as that Asperger's is a male dominated diagnosis, etc., and bigger pieces of support in terms of lateralisation. For example, language, in both sexes, is lateralised to the left hemisphere (see Broca's and Wernicke's areas). Irrespective of gender, I have found that people who show typical left brain traits in their behaviour (i.e., concerned with logic, system, sterile, emotionally cold, etc.) are Virgo Asc with mercury in first house.
Remember also that brain hemisphere is functional by contralaterality, meaning that the left eye projects to the right hemisphere and right eye to the left hemisphere. These projections are modulated by the corpus callosum, which facilitates hemispheric specialisation by allowing interhemispheric communication. Thus, if there was no communication between the hemispheres, more than likely we would have evolved with a language centre on each hemisphere rather than on just one. Children who have had a hemisphere removed end up developing all the centres that would be usually spread between two, in one.
I actually do not think that it is realistic to attribute hemispheres to planets because a hemispheres is only functional as a sum of its parts and each part can be said to correlate with a particular planetary resonance. For example, each hemisphere has grey matter, for processing, which contains cell bodies, and white matter for signal transduction, which contains only axons. Grey matter has six layers, some are excitatory some are inhibitory. Each cell body has a nucleus which holds the vitality of the cell: all the genetic information. And genetic information is again broken down into molecular significators, just like astrology, such that you get key players (i.e., planets): nucleobases of which there are only four (adenine, cytosine, guanine, thymine) but interact with each other in DNA strands within each person in a way that is as unique as, supposedly, an astrological chart. Just like the solar system is fast moving, so we do not get the same astrological profiles as the sun will never again be in the same place it was yesterday, so we never get a human with the same DNA as one that lived before.
But back to hemispheres, each hemisphere has lobes which are specialised. For instance, the frontal lobes are thought to be responsible for personality, motivation to do something, or lack of motivation in depression is correlated with decreased circulation in the fronal cortex. Also they are responsible for inhibition, such that people who have had frontal injury do obscene things in public. Frontal lobes are often called the executive centre of the brain because they also control memory and attention (both which may be seen as modulated by motivation). Being such an important part of maturity, I guess you could attribute them to Saturn. But then being such an important part of individuality, you may attribute them to the whole chart.
I dont know. But to me, asking which planet rules a hemisphere is like asking which planet rules humans (in general). In any event, given the evidence, I cant see it being as simple as the OP suggests.