The Rocket Experiment Demonstration of a Soft X-ray Polarimeter (REDSoX Polarimeter) is a sounding rocket instrument that can make the first measurement of the linear X-ray polarization of an extragalactic source in the 0.2-0.8 keV band as low as 10%. We employ multilayer-coated mirrors as Bragg reflectors at the Brewster angle. By matching the dispersion of a spectrometer using replicated optics from MSFC and critical angle transmission gratings from MIT to three laterally graded multilayer mirrors (LGMLs), we achieve polarization modulation factors over 90%. We present a novel arrangement of gratings, designed optimally for the purpose of polarimetry with a converging beam. The entrance aperture is divided into six equal sectors; pairs of blazed gratings from opposite sectors are oriented to disperse to the same LGML. The LGML position angles are 120 degrees to each other. CCD detectors then measure the intensities of the dispersed spectra after reflection and polarizing by the LGMLs, giving the three Stokes parameters needed to determine a source's linear polarization fraction and orientation. A current grant is funding further development to improve the LGMLs. Sample gratings for the project have been fabricated at MIT and the development team continues to improve them under separate funding. Our technological approach is the basis for a possible orbital mission.