We show that five nonnegativity properties of polynomials coincide when restricted to polynomials of the form $x_{1,\pi(1)}\cdots x_{n,\pi(n)} - x_{1,\sigma(1)}\cdots x_{n,\sigma(n)}$, where $\pi$ and $\sigma$ are permutations in $S_n$. In particular, we show that each of these properties may be used to characterize the Bruhat order on $S_n$.