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Birefringence, from the very essence of the word itself, refers to the splitting of light rays into two parts. In natural birefringent materials, this splitting is a beautiful phenomenon, resulting in the perception of a double image. In optical metamaterials, birefringence is often an unwanted side effect of forcing a device designed through transformation optics to operate in dielectrics. One polarization is usually implemented in dielectrics, and the other is sacrificed. Here we show, with techniques beyond transformation optics, that this need not be the case, that both polarizations can be controlled to perform useful tasks in dielectrics, and that rays, at all incident angles, can even follow different trajectories through a device and emerge together as if the birefringence did not exist at all. A number of examples are shown, including a combination Maxwell fisheye/Luneburg lens that performs a useful task and is achievable with current fabrication materials.