Cerebral lateralization
The two halves of the brain are not duplicates — they specialise. In most people the left hemisphere runs language and speech, while the right is better at spatial layout, faces, and seeing things whole. Split-brain patients made this visible in the cleanest possible way, because you could send a word or a picture to one hemisphere alone and watch which half could deal with it.
The functional specialisation of the two cerebral hemispheres. In roughly 95% of right-handers the left hemisphere is dominant for language and speech; the right hemisphere specialises in spatial processing, face recognition, and holistic perception. Split-brain testing demonstrated this directly: a word flashed to the left hemisphere could be spoken, while the same word flashed to the right hemisphere could not — though the left hand, controlled by the right hemisphere, could point to the object it named.
Lateralisation is what turned the split-brain experiments from a clinical curiosity into a map. Because the disconnected hemispheres could be tested one at a time, Sperry and Gazzaniga could show that speech lived on the left and that the mute right hemisphere was not stupid but specialised — better, often, at spatial and face tasks. The asymmetry is the reason the disconnection produces such strange scenes: the half that can talk does not have direct access to what the other half saw, so when the two are shown different things at once, the talking half is left explaining a hand it did not direct — the opening that Gazzaniga's interpreter walks through.