Plasmonic devices designed in three dimensions enable careful tuning of optical responses for control of complex electromagnetic interactions on the nanoscale. Probing the fundamental characteristics of the constituent nanoparticle building blocks is, however, often constrained by diffraction-limited spatial resolution in optical spectroscopy. Electron microscopy techniques, including electron energy loss spectroscopy (EELS), have recently been developed to image surface plasmon resonances qualitatively at the nanoscale in three dimensions using tomographic reconstruction techniques. Here, we present an experimental realization of a distinct method that uses direct analysis of modal surface charge distributions to reconstruct quantitatively the three-dimensional eigenmodes of a silver right bipyramid on a metal oxide substrate. This eigenmode tomography removes ambiguity in two-dimensional imaging of spatially-localized plasmonic resonances, reveals substrate-induced mode degeneracy breaking in the bipyramid, and enables EELS for the analysis not of a particular electron-induced response but of the underlying geometric modes characteristic of particle surface plasmons.