We introduce a calculus for the class A S M ( X ) of direct images of semi-meromorphic currents on a reduded analytic space X , that extends the classical calculus due to Coleff, Herrera and Passare. Our main result is that each element in this class acts as a kind of multiplication on the sheaf 𝒫ℳ X of pseudomeromorphic currents on X . We also prove that A S M ( X ) as well as 𝒫ℳ X and certain subsheaves are closed under the action of holomorphic differential operators and interior multiplication by holomorphic vector fields.