The second-order moments method is a standard method to characterize laser beams. All beams are described
by a beam matrix, with specific mathematical properties. The geometrical classification of these beams is
based on their matrix structure, or symmetry, rather than on the symmetry of beams' cross sections in freespace
propagation. Accordingly, the beams can be stigmatic, simple astigmatic, or general astigmatic. On the
other hand, at propagation through ABCD-type optical systems, some intrinsic properties of the beams
remain invariant. Two independent invariant quantities define two classes and four types of families of
beams, providing the intrinsic classification of beams, irrespective of their geometrical classification. The
paper reveals the intrinsic classification of the general astigmatic beams, by determining the intrinsic class
and family each general astigmatic beam matrix belongs to. First, we summarize previous results which are
important to understand and to obtain the new results of this paper. This includes the beam description using
two more mathematical models, one using the same beam matrix with a different order of the elements, and
the other being the gaussian-Schell-model. Then we show what kind of information on the invariants we can
retrieve from the mathematical properties of the different quantities used in the three beam descriptions.
Finally we analyze all possible non-degenerate matrices representing general astigmatic beams (ten in all) and
apply the information retrieved as described above for each of them. The final result is a list of all ten
matrices representing general astigmatic beams and their intrinsic classification.