  Primary cohomology operations, i.e., elements of the Steenrod algebra, are
given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps
(before taking homotopy classes) form the topological version of the Steenrod
algebra. Composition of such maps is strictly linear in one variable and linear
up to coherent homotopy in the other variable. To describe this structure, we
introduce a hierarchy of higher distributivity laws, and prove that the
topological Steenrod algebra satisfies all of them. We show that the higher
distributivity laws are homotopy invariant in a suitable sense. As an
application of $2$-distributivity, we provide a new construction of a
derivation of degree $-2$ of the mod $2$ Steenrod algebra.
