The steam-assisted gravity-drainage (SAGD) method has emerged as among the leading methods of enhanced oil recovery and is predicated on the injection of steam within the wellbore followed by extraction of emulsions of viscous oil and water. The emulsions are stabilized by endogenous surfactants, necessitating extensive processing such as addition of chemical de-emulsifiers and slow gravity-based separation methods. Here, we show that a hierarchically textured membrane exhibiting orthogonal wettability, specifically, superoleophilic but superhydrophobic behavior, allows for effective separation of the water and viscous oil fractions of SAGD emulsions. The membrane is constructed by integrating ZnO nanotetrapods onto stainless steel meshes using a conformal amorphous SiO2 layer and is both mechanically resilient and thermally robust. The intrinsic surface energy characteristics of the ZnO tetrapods as well as their three-dimensional texture when arrayed atop the stainless steel mesh substrates contribute to the observed differential wettability between water and oil. Water content in permeated bitumen is reduced to as low as 0.69 vol % through a single-pass filtration step with the further advantage of eliminating silt particles. The permeation temperature and water content are tunable based on modulation of the mesh size and ZnO loading. The membranes allow for operation at SAGD temperatures in excess of 130 degrees C, thereby enabling the thermal disruption of hierarchical emulsions. The membrane-based separation of SAGD emulsions under realistic process conditions paves the way for entirely new process designs for recovering dry viscous oil.