In contrast with what happens for two-dimensional polarization states, defined as those whose electric field fluctuates in a fixed plane, which can readily be represented by means of the Poincaré sphere, the complete description of general three-dimensional polarization states involves nine measurable parameters, called the generalized Stokes parameters, so that the generalized Poincaré object takes the complicated form of an eight-dimensional quadric hypersurface. In this work, the geometric representation of general polarization states, described by means of a simple polarization object constituted by the combination of an ellipsoid and a vector, is interpreted in terms of the intrinsic Stokes parameters, which allows for a complete and systematic classification of polarization states in terms of meaningful rotationally invariant descriptors.