The Tharsis region is the largest volcanic complex on Mars and in the Solar System. Young lava flows cover its surface (from the Amazonian period, less than 3 billion years ago) but its growth started during the Noachian era (more than 3.7 billion years ago). Its position has induced a reorientation of the planet with respect to its spin axis (true polar wander, TPW), which is responsible for the present equatorial position of the volcanic province. It has been suggested that the Tharsis load on the lithosphere influenced the orientation of the Noachian/Early Hesperian (more than 3.5 billion years ago) valley networks(1) and therefore that most of the topography of Tharsis was completed before fluvial incision. Here we calculate the rotational figure of Mars (that is, its equilibrium shape) and its surface topography before Tharsis formed, when the spin axis of the planet was controlled by the difference in elevation between the northern and southern hemispheres (hemispheric dichotomy). We show that the observed directions of valley networks are also consistent with topographic gradients in this configuration and thus do not require the presence of the Tharsis load. Furthermore, the distribution of the valleys along a small circle tilted with respect to the equator is found to correspond to a southern-hemisphere latitudinal band in the pre-TPW geographical frame. Preferential accumulation of ice or water in a south tropical band is predicted by climate model simulations of early Mars applied to the pre-TPW topography. A late growth of Tharsis, contemporaneous with valley incision, has several implications for the early geological history of Mars, including the existence of glacial environments near the locations of the pre-TPW poles of rotation, and a possible link between volcanic outgassing from Tharsis and the stability of liquid water at the surface of Mars.