  We prove that the set of orthogonal separable coordinates on an arbitrary
(pseudo-)Riemannian manifold carries a natural structure of a projective
variety, equipped with an action of the isometry group. This leads us to
propose a new, algebraic geometric approach to the classification of orthogonal
separable coordinates by studying the structure of this variety. We give an
example where this approach reveals unexpected structure in the well known
classification and pose a number of problems arising naturally in this context.
