We analyze the mid-infrared (MIR) spectra of ultraluminous infrared galaxies (ULIRGs) observed with the Spitzer Space Telescope's Infrared Spectrograph. Dust emission dominates the MIR spectra of ULIRGs, and the reprocessed radiation that emerges is independent of the underlying heating spectrum. Instead, the resulting emission depends sensitively on the geometric distribution of the dust, which we diagnose with comparisons of numerical simulations of radiative transfer. Quantifying the silicate emission and absorption features that appear near 10 and 18um requires a reliable determination of the continuum, and we demonstrate that including a measurement of the continuum at intermediate wavelength (between the features) produces accurate results at all optical depths. With high-quality spectra, we successfully use the silicate features to constrain the dust chemistry. The observations of the ULIRGs and local sightlines require dust that has a relatively high 18/10um absorption ratio of the silicate features (around 0.5). Specifically, the cold dust of Ossenkopf et al. (1992) is consistent with the observations, while other dust models are not. We use the silicate feature strengths to identify two families of ULIRGs, in which the dust distributions are fundamentally different. Optical spectral classifications are related to these families. In ULIRGs that harbor an active galactic nucleus, the spectrally broad lines are detected only when the nuclear surroundings are clumpy. In contrast, the sources of lower ionization optical spectra are deeply embedded in smooth distributions of optically thick dust.

38 pages, 11 figures; to appear in ApJ v679 (May 20)