Omnidirectional reflection (ODR) from birefringent multilayers consisting of uniaxial layers in the principal-axis system alternating with isotropic layers is further analysed. Large domains of layer refractive indices are considered. Projected band structures are shown comparatively for positive and negative birefringent multilayers. Diagrams with the pairs of layer refractive indices at which the lowest ODR band does not exist or has certain constant values are given. Exact analytic relations are given for the coordinates of the Brewster crossing point in case of isotropic and birefringent multilayers. Results show that the birefringence may enhance the ODR. Wider ODR bands may be achieved with positive birefringent multilayers. Even wider ODR bands result by chirping down the layer period. Results presented may be useful not only in the domain of one-dimensional photonic bandgap structures but also in other schemes that include birefringent multilayers.