The cerebral hemispheres are involved with inputs and outputs from the contralateral side of the body
Damage to neocortex causes problems on the opposite side
In patients with epilepsy, surgeons occasionally cut corpus callosum to relieve seizures. Flash different pictures in each eye, patients could describe what they saw with right eye, but not left, but could pick out object - example: Heart = ART.
The cerebellum is involved with the control of movement on the ipsilateral side of the body.
Damage to the cerebellum causes motor deficits on the same side.