The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup-$i$ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. This correspondence allows us to interpret the coproducts geometrically in terms of zonotopal tilings, understand all possible choices of coproducts, and give conceptual proofs of their properties.