The balanced canonical form and parametrization of Ober for the case of SISO stable systems are extended to block-balanced canonical forms and related input-normal forms and parametrizations. They form an overlapping atlas of parametrizations of the manifold of stable SISO systems of given order. This extends the usefulness of these parametrizations, e.g., in gradient algorithms for system identification. As an implication of our construction it follows that each of the subsets of the parametrization of [R. Ober, Internat. J. Control, 46 (1987), pp. 683-670] corresponding to a choice for the structural indices is in fact an imbedded submanifold of the manifold of stable SISO systems of fixed order.