  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.
