In addition to the double dimensional reduction procedure that employs world-volume Killing symmetries of p-brane supergravity solutions and acts diagonally on a plot of p versus space-time dimension D, there exists a second procedure of “vertical” reduction. This reduces the transverse-space dimension via an integral that superposes solutions to the underlying Laplace equation. We show that vertical reduction is also closely related to the recently introduced notion of intersecting p-branes. We illustrate this with examples, and also construct a new D = 11 solution describing four intersecting membranes, which preserves 116 of the assymetry. Given the two reduction schemes plus duality transformations at special points of the scalar modulus space, one may relate most of the p-brane solutions of relevance to superstring theory. We argue that the maximum classifying duality symmetry for this purpose is the Weyl group of the corresponding Cremmer-Julia supergravity symmetry Er(+r). We also discuss a separate class of duality-invariant p-branes with p = D − 3.