A spectrometer based on a Sagnac interferometer, where one of the mirrors is replaced by a transmission grating, is introduced. Since the action of a transmission grating is reversible, both directions experience the same diffraction at a given wavelength. At the output, the crossed wavefronts are imaged onto a camera, where their Fizeau fringe pattern is recorded. Each spectral element produces a unique spatial frequency, hence the Fourier transform of the recorded interferogram contains the spectrum. Since the grating is tuned to place zero spatial frequency at a selected wavelength, the adjoining spectrum is heterodyned with respect to this wavelength. This spectrum can then be discriminated at a high spectral resolution from relatively low spatial frequencies. The spectrometer can be designed without moving parts for a relatively narrow spectral range or with a rotatable grating. The latter version bears the potential to be calibrated without a calibrated light source.