Continuous-trace C*-algebras A with spectrum T can be characterized as those algebras which are locally Monta equivalent to C0(T). The Dixmier-Douady class δ(A) is an element of the Čech cohomology group Ȟ3(T, ℤ) and is the obstruction to building a global equivalence from the local equivalences. Here we shall be concerned with systems (A, G, α) which are locally Monta equivalent to their spectral system (C0(T),G, τ), in which G acts on the spectrum T of A via the action induced by α. Such systems include locally unitary actions as well as N-principal systems. Our new Dixmier-Douady class δ (A, G, α) will be the obstruction to piecing the local equivalences together to form a Monta equivalence of (A, G, α) with its spectral system. Our first main theorem is that two systems (A, G, α) and (B, G, β) are Monta equivalent if and only if δ (A, G, α) = δ (B, G, β). In our second main theorem, we give a detailed formula for δ (A ⋊α G) when (A, G, α) is N-principal.