A new technique for deriving the degree of linear polarization from limb-scattered sunlight radiance measurements is presented and applied to Odin/OSIRIS. In this technique, the measured OSIRIS grating efficiency spectrum is fit to OSIRIS limb spectra using the differential optical absorption spectroscopy technique and the resultant fitting constant is shown to represent the difference in linear polarization between two tangent heights. The validity of this technique is confirmed using simulations from a vector limb radiative transfer model. OSIRIS scans of the difference in linear polarization from October 2003 are shown to be very sensitive to stratospheric and upper tropospheric aerosols, including background sulphates, clouds, and polar stratospheric clouds.