We present an analysis of a sample of 69 local obscured Swift/Burst Alert Telescope active galactic nuclei (AGNs) with X-ray spectra from NuSTAR and infrared (IR) spectral energy distributions from Herschel and WISE. We combine this X-ray and IR phenomenological modeling and find a significant correlation between reflected hard X-ray emission and IR AGN emission, with suggestive indications that this correlation may be stronger than the one between intrinsic hard X-ray and IR emissions. This relation between the IR and reflected X-ray emission suggests that both are the result of the processing of intrinsic emission from the corona and accretion disk by the same structure. We explore the resulting implications on the underlying distribution of covering fraction for all AGNs, by generating mock observables for the reflection parameter and IR luminosity ratio using empirical relations found for the covering fraction with each quantity. We find that the observed distributions of the reflection parameter and IR-to-X-ray ratio are reproduced with broad distributions centered around covering fractions of at least similar to 40%-50%, whereas narrower distributions match our observations only when centered around covering fractions of similar to 70%-80%. Our results are consistent with both independent estimates of the covering fractions of individual objects and the typical covering fraction obtained on the basis of obscured fractions for samples of AGNs. These results suggest that the level of reprocessing in AGNs, including X-ray reflection, is related in a relatively straightforward way to the geometry of the obscuring material.