The inferred black hole in the Galactic center spans the largest angle on the sky among all known black holes. Forthcoming observational programs plan to localize or potentially resolve the image of Sgr A* to an exquisite precision, comparable to the scale of the black hole horizon. Here we show that the location of the image centroid of Sgr A* should depend on observing frequency because of relativistic and radiative transfer effects. The same effects introduce a generic dependence of the source polarization on frequency. Future detection of the predicted centroid shift and the polarization dependence on frequency can be used to determine the unknown black hole spin and verify the validity of General Relativity.