High-resolution long-slit spectra obtained with the Phoenix spectrograph on Gemini South provide our most accurate probe of the three-dimensional structure of the Homunculus Nebula around η Carinae. The new near-infrared spectra dramatically confirm the double-shell structure inferred previously from thermal dust emission, resolving the nebula into a very thin outer shell seen in H₂ and a warmer, thicker inner layer seen in [Fe II]. The remarkably thin and uniform H ₂ skin hints that the most important mass loss during the 19th century eruption had a very short duration of 5 yr. H₂ emission traces the majority of the more than 10 M of material in the nebula and has an average density of order n_H 10^6.5 cm^-3. This emission, in turn, yields our first definitive picture of the exact shape of the nebula, plus a distance of 2350 ± 50 pc and an inclination angle of ~41°. The distribution of the H₂ emission provides the first measure of the latitude dependence of the speed, mass loss, and kinetic energy associated with η Car's 19th century explosion. Almost 75% of the total mass and more than 90% of the kinetic energy in the ejecta were released at high latitudes between 45° and the polar axis. This rules out a model for the bipolar shape in which an otherwise spherical explosion was pinched at the waist by a circumstellar torus. Instead, most of the mass appears to have been directed poleward by the explosion itself. H₂ emission also provides our first reliable picture of the critical innermost waist of the Homunculus, yielding clues to the observed morphology of the core and the more extended equatorial debris.