Sgr A East appears to be a single, mixed-morphology 10,000 yr old supernova remnant (SNR) at the Galactic center. It also appears to belong to a class of remnants that have been observed and detected at 1720 MHz, the transition frequency of OH maser emission. However, if the EGRET source 3EG J1746-2852, coincident with the Galactic center, is itself associated with this object, it would endow it with a γ-ray luminosity almost 2 orders of magnitude greater than that of the other EGRET-detected SNRs. We here reconsider the viability of a pion-production mechanism as the source of the broadband emission observed from Sgr A East, and show that what connects these objects—and ultimately also accounts for their different γ-ray emissivity—is the very important interaction between the expanding SNR shell and the surrounding molecular cloud environment. The singularly high γ-ray luminosity of Sgr A East, as well as its unusually steep radio spectral index, can thereby be attributed to the high-density (n_H = 10³ cm^-3), strongly magnetized (B ~ 0.18 mG) environment in which it is located.