We searched for positional brain surface asymmetries measured as displacements between corresponding vertex pairs in relation to a mid-sagittal plane in Magnetic Resonance (MR) images of the brains of 223 humans and 70 chimpanzees. In humans deviations from symmetry were observed: 1) a Torque pattern comprising right-frontal and left-occipital “petalia” together with downward and rightward “bending” of the occipital extremity, 2) leftward displacement of the anterior temporal lobe and the anterior and central segments of superior temporal sulcus (STS), and 3) posteriorly in the position of left occipito-temporal surface accompanied by a clockwise rotation of the left Sylvian Fissure around the left-right axis. None of these asymmetries was detected in the chimpanzee, nor was associated with a sex difference. However, 4) an area of cortex with its long axis parallel to the olfactory tract in the orbital surface of the frontal lobe was found in humans to be located higher on the left in females and higher on the right in males. In addition whereas the two hemispheres of the chimpanzee brain are equal in extent in each of the three dimensions of space, in the human brain the left hemisphere is longer (p = 3.6e-12), and of less height (p = 1.9e-3), but equal in width compared to the right. Thus the asymmetries in the human brain are potential correlates of the evolution of the faculty of language.