The Galaxy’s supermassive black hole, Sgr A*, produces an outburst of infrared (IR) radiation about once every 6 h, sometimes accompanied by an even more energetic flurry of X-rays. It is rather clear now that the near-IR (NIR) photons are produced by non-thermal synchrotron processes, but we still do not completely understand where or why these flares originate, nor exactly how the X-rays are emitted. Circumstantial evidence suggests that the power-law electrons radiating the IR light may be partially cooled, allowing for the possibility that their distribution should be more accurately described by a broken power law with a (‘cooling break’) transition frequency. In addition, the emission region (energized by an as yet unidentified instability) appears to be rather compact, possibly restricted to the inner edge of the accretion disc. In that case, the X-ray outburst may itself be due to synchrotron processes by the most energetic particles in this population. In this paper, we examine several key features of this proposal, producing relativistically correct polarimetric images of Sgr A*’s NIR and X-ray flare emission, in order to determine (1) whether the measured NIR polarization fraction is consistent with this geometry and (2) whether the predicted X-ray to NIR peak fluxes are confirmed by the currently available multiwavelength observations. We also calculate the X-ray polarization fraction and position angle (relative to that of the NIR photons) in anticipation of such measurements in the coming years. We show that whereas the polarization fraction and position angle of the X-rays are similar to those of the NIR component for synchrotron-cooled emission, these quantities are measurably different when the X-rays emerge from a scattering medium. It is clear, therefore, that the development of X-ray polarimetry will represent a major new tool for studying the space–time near supermassive black holes.