The bipolar reflection nebula OH 231.8+4.2 (OH 0739-14), centred on the Mira variable star QX Pup, has been mapped at Ks in linear polarization using the ESO SOFI near-infrared multi-mode instrument. The large scale linear polarization features agree with previous studies, validating the polarimetric mode of the instrument. However the high spatial resolution of the polarization map reveals considerable structure, some of which finds correspondence in the surface brightness images. The core is crossed by low polarization filaments; the northern lobe reveals an axial finger and a rim of very high linear polarization (upto 80%) whilst the southern lobe shows less polarization contrast. There is a trend in polarization perpendicular to the bipolar axis in both lobes. The individual linear polarization features most probably arise from localised dust structures with distinct orientations leading to differing polarization through single scattering. A single scattering geometric model, consisting of two lobes of ellipsoidal section and a central dense disk, is considered which can explain the general features of the intensity and polarization images.