The standard picture of galaxy formation motivates the decomposition of the Galaxy into stellar populations and chemical abundances. To test this idea, we construct a Gaussian mixture model (GMM) for stars in the Solar neighborhood, assessing data in the space of their measured velocities and iron abundances (i.e. an augmented Toomre diagram). We compare results for the Gaia-APOGEE crossmatch catalog of the solar neighborhood with those from a suite of synthetic Gaia-APOGEE crossmatches constructed from high-resolution, cosmological-hydrodynamical simulations of Milky-Way-mass galaxies. We find that in both the synthetic and real data, the best-fit GMM uses five independent components whose properties resemble the populations predicted by galaxy formation theory. The best-fit model matches our physical intuitions about the origin of Solar-neighborhood stars and their spatial, kinematic, and chemical abundance distributions. Quantitatively, we get new insights, as the optimal decomposition features five components: two analogous to the thin disk and the halo, but instead of a single counterpart to the thick disk, there are three intermediate components. We demonstrate how the model trained on the real data, once interpreted using the mock catalog, can subsequently be used to determine interesting subpopulations of stars from other surveys with more complex selection functions.