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?

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?

Edit: When I say "sheaf of twisted differential operators" I just mean a quasicoherent sheaf of filtered associative algebras $\mathscr{A}$ on $X$ together with an isomorphism $\text{gr } \mathscr{A} \cong \text{Sym } \mathscr{T}_X$. The answer probably doesn't involve any difficult computations but I'm pretty murky about what's happening on the cotangent bundle. Any general remarks in that direction, especially from an algebraic perspective, would be helpful.