Loss of water to space via neutral hydrogen escape has been an important process throughout Martian history. Contemporary loss rates can be constrained through observations of the extended neutral hydrogen atmosphere of Mars in scattered sunlight at 121.6 nm. Historically, such observations have been interpreted with coupled density and radiative transfer models, inferring escape fluxes from brightness profiles gathered by flybys, orbiters, and telescope observations. Here we demonstrate that the spherical symmetry assumed by prior analyses cannot reproduce observations by the Imaging Ultraviolet Spectrograph (IUVS) on the Mars Atmosphere and Volatile EvolutioN (MAVEN) mission. We present unique observations of the Mars H corona to large radial distances and mapping results from initial MAVEN science at Mars. These observations represent the first detection of three-dimensional structure in the H corona of Mars, with implications for understanding the atmosphere today and the loss of H to space throughout Martian history.