Some years ago we developed an automatized reflectometric method to measure the surface azimuthal anchoring energy of nematic liquid crystals on an optically isotropic substrate. This method provides a high accuracy and sensitivity but requires the use of wedge glass plates and a sufficiently high anisotropy of the intensity reflectivity coefficients. This latter condition restricts greatly the number of possible substrates that can be investigated with this technique. Here we develop a new reflectometric method which offers comparable or better accuracy and sensitivity but does not require wedge plates and high anisotropy of the reflectivity coefficients. The method is fully automated and provides a direct measurement of the azimuthal director angle. The experimental procedure exploits the dependence of the reflectivity tensor on the surface director orientation. The measurement of the azimuthal angle does not require any knowledge of the optical parameters of the nematic material and of the optically isotropic substrate, and provides an absolute accuracy better than 0.2( degrees ) in the whole range 0-360( degrees ) and a sensitivity better than 0.1( degrees ). This reflectometric method can be also used with weakly anisotropic substrates as well as thin rubbed polymeric layers. In this latter case, the effective uncertainty in the measurement of the director azimuthal angle depends on the substrate anisotropy. A simple and direct experimental procedure to estimate this uncertainty is proposed.