Transverse-periodic-oriented nematic liquid crystals (LCs) are a special type of optical axis grating that are capable of very high efficiency diffraction (theoretically, 100%) in thin layers of materials with thickness comparable to the radiation wavelength. In particular, they fully diffract linearly polarized input beam into circularly polarized +1st and -1st diffraction orders. We experimentally demonstrate switching between diffraction orders of such gratings when the polarization of the incident beam changes from right-circular to left-circular and vice versa with the aid of an electrically controlled LC phase retarder. Such a setup in which the diffraction efficiency and direction are controlled externally, without application of an electric field to the transverse-periodic grating, provides additional control opportunities and does not compromise the quality of the grating. The grating used in the experiment was 1.5 microm thick and had a period of 4 microm. The contrast ratio of switching between the +1st and -1st orders was as high as 267:1 for a He-Ne laser beam with a switching time of 6.6 ms.