Far-infrared polarimetry is a well known tool to measure the poloidal field Bp in terms of the density-weighted line-of-sight integral of Bp|| which is proportional to the Faraday rotation angle of the plane of polarization. The instrument at JET uses a wavelength of 195mm. It comprises four vertical and four lateral channels (chords) with the same probing beams utilized for interferometry as well. It has been operational for many years. This 'traditional' polarimetry provides important data to calculate the q-profile, but does not exploit the method completely. The general polarization state involves two parameters: the angle of the plane of polarization Y and the ellipticity. The latter results from the Cotton-Mouton phase shift which is proportional to the line integral of ne if the toroidal field is largely constant along the line of sight. Thus, interferometry-independent line-integrated density signals, which are free from fringe jumps, can be obtained for vertical chords. In fact, this 'dual potential' of polarimetry has long been known theoretically, and exploitation of the Cotton-Mouton effect as a density diagnostic has been suggested for ITER. Now the present paper reports tangible and very encouraging experimental results from a large tokamak. However, its main topic are the new experimental and theoretical methods developed to overcome two major obstacles which may so far have prevented measurements like this on a routine basis.