SU(2) gadgets for counterpropagating polarization states

It is well known that gadgets consisting of simple combinations of fixed linear retarders can be used to generate any SU(2) transformation of a polarization state. However, photons propagating through such a gadget in reverse will in general undergo a completely different transformation. We discuss how to engineer the relation between the transformations in the two different propagation directions, and show how they can be made to be identical, the adjoint or transpose of each other, or in the most general case two completely independent, arbitrary transformations. This is made possible through the use of fixed Faraday rotators, which allow for selective engineering of the symmetry properties of a gadget under change of propagation direction. Our work opens up new possibilities for controlling the polarization of light inside Sagnac interferometers and in other common-path geometries. In particular it can greatly simplify optical implementations of the quantum SWITCH and enable the realisation of new processes such as the recently proposed quantum time flip.