The double spherical harmonics method has hitherto been considered applicable only if two conditions are fulfilled: (a) the directions of the assumed angular discontinuities could be assigned a priori by symmetry, and (b) at every boundary surface, these directions lie in a tangent plane. In this paper, the DP-0 approximation is reformulated to cover all situations; for some new ones, the accuracy is predictably better than that from the P-1 approximation. The directions of the angular discontinuity now have to be calculated, but a straightforward generalization of the usual derivation fails to furnish such a procedure. This generalization also retains a former uncertainty about how much of a delta function is on each side of the angular discontinuity. A variational principle is used as one way to settle both of these issues. Anisotropic scattering is allowed. Remarkably, the completed formulation differs from ordinary diffusion theory, including boundary conditions, only in the definition of the diffusion coefficient. A standard diffusion theory calculation can accommodate whichever type of theory is preferred in any particular region of a single problem; the adjustment is made merely by assigning the corresponding diffusion coefficient there. The variational principle implies that the delta function should be divided evenly into two parts at the discontinuity. However, this division is not a mathematically inherent property of the delta function, as sometimes thought, and other formulations for spherical and cylindrical geometries have placed the delta function entirely in the forward hemisphere of directions. Examples showing superior accuracy and physical interpretations supported that choice, but they are matched here by other examples favoring different divisions.