For the past decade, the Lyman-alpha detectors on board National Aeronautics and Space Administration's Two Wide-angle Imaging Neutral-atom Spectrometers (TWINS) mission have obtained routine measurements of solar Lyman-α photons (121.6 nm) resonantly scattered by atomic hydrogen (H) in the terrestrial exosphere. These data have been used to derive global three-dimensional (3-D) models of exospheric H density beyond 3 RE, which are needed to understand various aspects of aeronomy and heliophysics, such as atmospheric chemistry and energetics, magnetospheric energy dissipation, ion-neutral coupling, and atmospheric evolution through gravitational escape. These empirical distributions are obtained through parametric fitting of assumed functional forms that have little observational justification, thus limiting confidence in conclusions drawn from analysis of the resulting exospheric structure. In this work, we present a new means of global 3-D reconstruction of exospheric H density through tomographic inversion of the scattered H Lyman-α emission. Our approach avoids the conventional dependence on ad hoc parametric formulations and, based on the case studies reported here, appears to enable a more accurate characterization of the global structure of the H density in the outer exosphere. We evaluate the bounds of technique feasibility using simulated TWINS data and report new geophysical insights gained from applying this promising new approach to an example of actual TWINS data.