  In recent work, the authors used an order lowering operator $\nabla$,
introduced by Stanley, to prove the strong Sperner property for the weak Bruhat
order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt
interpreted $\nabla$ as a differential operator on Schubert polynomials and
used this to prove a new identity for Schubert polynomials and a determinant
conjecture of Stanley. In this paper we study a raising operator $\Delta$ for
the \emph{strong} Bruhat order, which is in many ways dual to $\nabla$. We
prove a Schubert identity dual to that of Hamaker et al. and derive formulas
for counting weighted paths in the Hasse diagrams of the strong order which
agree with path counting formulas for the weak order. We also show that powers
of $\nabla$ and $\Delta$ have the same Smith normal forms, which we describe
explicitly, answering a question of Stanley.
