Liquid crystal (LC) devices are presented which convert linearly polarized light into linearly polarized light with axial symmetry. Their symmetry axis is determined by the propagation axis of the light beam. Such light fields can be characterized by an integer P, which we call the polarization order number. Our LC devices are shown to generate P = -2, -1, 1 and 2 fields. Devices that generate P = 1 fields act as azimuthal (A) or radial (R) polarizers or analyzers. Our circularly symmetric polarizers can be used as polarization axis finders or, when two of them are combined, as versatile tools for investigating birefringent, dichroic or optically active materials.