  We prove that twisted versions of Schubert polynomials defined by
$\widetilde{\mathfrak S}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}$ and
$\widetilde{\mathfrak S}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak S}_w$
are monomial positive and give a combinatorial formula for their coefficients.
In doing so, we reprove and extend a previous result about positivity of skew
divided difference operators and show how it implies the Pieri rule for
Schubert polynomials. We also give positive formulas for double versions of the
$\widetilde{\mathfrak S}_w$ as well as their localizations.
