The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\mathfrak{G}^{(β)}_w(x;y)$ as a sum of products $\mathfrak{G}^{(β)}_v(x)\mathfrak{G}^{(β)}_u(y)$ of single Grothendieck polynomials. Combinatorially, this identity suggests the existence of a weight-preserving bijection between certain families of diagrams called pipe dreams. In this paper, we provide such a bijection using an algorithm called pipe dream rectification. In turn, rectification is built from a new class of flow operators which themselves exhibit a surprising symmetry. Finally, we examine other applications of rectification including an insertion algorithm on pipe dreams which recovers a variant of the dual RSK correspondence.