It has been argued that F-R I radio galaxies and BL Lac objects are intrinsically the same. The strong emission seen in BL Lac objects may be the result of beaming of radiation from the galactic nucleus. We see a BL Lac object when our line of sight lies within the beam. If this is correct, then F-R I radio galaxies also have very luminous beams of radiation, which are directed away from our line of sight. Many F-R I galaxies are found in the centers of clusters of galaxies, and many clusters contain cooling flows. The high gas densities in these regions imply small but significant electron scattering optical depths. We show that the cooling flow gas will scatter the beamed emission and make it visible even when the beams are directed away from our line of sight. We derive the surface brightness of the scattered light for beams of polarized and unpolarized radiation and determine the polarization properties of the scattered light. Assuming typical angles for the beams, a typical BL Lac spectrum and apparent luminosity, and typical cooling flow properties, we derive the expected scattered light surface brightness from radio to X-ray wavelengths. The surface brightness of scattered light is compared to that of the direct emission from the galaxy and its environment, assuming typical cD galaxy properties. We find that the scattered emission will be most easily detected at radio, infrared, and blue/UV wavelengths. Such observations could provide an important means of detecting the parent population of high-luminosity AGNs. We compare our models to recent observations of lobes of excess, polarized blue light in Cyg-A, and observations of lobes of UV light in the central regions of the cD galaxies in A1795 and A2597. These lobes are aligned with the radio axis. Our models predict scattered surface brightnesses in reasonable agreement with those observed. If these lobes are indeed scattered light, this would show that F-R I radio galaxies do possess beams of radiation with apparent luminosities similar to BL Lac objects, that these beams are double and that the radiation is actively beamed and not merely obscured by a torus of dust around the nucleus.