Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra $\text{gr } \mathscr{A}$ is identified with $\text{Sym } \mathscr{T}_X$, the sheaf of functions on the cotangent bundle $T^*X$.

Both of these algebras have a Poisson structure: $\text{gr } \mathscr{A}$ the one induced by the commutator, and $\text{Sym } \mathscr{T}_X$ the one arising from the canonical symplectic structure on $T^*X$. Can anyone explain to me why these Poisson structures coincide under the aforementioned isomorphism?