Let $X$ be your "favourite" kind of space, and let $\mathcal{D}_X$ be the (sheaf of) ring(s) of differential operators on $X$. What does the ring $\mathcal{D}_X$ tell us about $X$?

I know this might be too broad or naive a question, but the literature seems to be equally vast and scattered. Browsing the canonical references on rings of differential operators (Bjork, McConnell & Robson) and/or $\mathcal{D}$-modules (Bjork, Borel, Hotta et al), quite a bit is known in the forward direction, e.g. if $X$ is smooth over $\mathbb{C}$ then $\mathcal{D}_X$ is $\mathcal{O}_X$-quasicoherent and (left and right) noetherian; $\text{gr}\mathcal{D}_X\simeq \mathcal{O}_{T^*X}$; ...

In fact, one of the points of this question is also to have an idea for what kind of "favourite" spaces this is an approachable problem, and what is known out there in the literature. In particular,

If $\mathcal{D}_X\simeq \mathcal{D}_Y$ as rings, what does this tell us about the relation between $X$ and $Y$? That is, how much the isomorphism class of rings $\mathcal{D}_X$ is an interesting object to understand $X$? And, in a more Tannakian fashion, how much of $X$ can we recover from $\mathcal{D}_X$?

References most welcome. Please, re-tag if appropriate.