Combined neutron scattering and diffusion nuclear magnetic resonance experiments have been used to reveal significant interregional asymmetries (lateralization) in bovine brain hemispheres in terms of myelin arrangement and water dynamics at micron to atomic scales. Thicker myelin sheaths were found in the left hemisphere using neutron diffraction. 4.7 T dMRI and quasi-elastic neutron experiments highlighted significant differences in the properties of water dynamics in the two hemispheres. The results were interpreted in terms of hemisphere-dependent cellular composition (number of neurons, cell distribution, etc.) as well as specificity of neurological functions (such as preferential networking).