The \Cross Product" technique introduced by Even and Litman EL92] is extended into a fulldecompositiontheory enablinga unique (up to isomorphism)and polynomialfactorizationof layered interconnection networks (including most well-known networks) into a product of prime factors. A polynomialalgorithmis provided for checking whether a givenlayered interconnection network is isomorphic to a uniquely decomposible network into prime factors.