Let $X$ be a smooth variety and let $Y\subset X$ be a closed smooth subvariety. We have the local cohomology $H^i_Y(X)$, which becomes a MHS via the mixed cone construction, as explained in Section 5.5 from "Mixed hodge structures" by Peters and Steenbrink.

One also has the Thom isomorphism $H_Y^i(X)\cong H^{i-c}(Y)$, where $c$ is the codimension of $Y$ in $X$. $H^{i-c}(Y)$ is itself also equipped with a (canonical) MHS.

Now I wonder what the relation between these two MHS's is. I am particularly puzzled by this because the MHS on $H^{i-c}(Y)$ is canonical (i.e. independent of the embedding $Y\subset X$), while the cone construction is not canonical at all.

I have not been able to find a reference for this in the literature, and working out the details of the cone construction and combining it with the Thom isomorphism map proved quite intricate and I have not been successful with this analysis.

Duality of Mixed Hodge Structures of Algebraic Varieties, it is proved that there is a natural perfect pairing $H^i_Y(Y,\mathbb Q)\times H^{2n-i}(Y,\mathbb Q)\to \mathbb Q$ which induces duality of mixed $\mathbb Q$-Hodge structures. Together with the standard pairing $H^{i-2c}(Y,\mathbb Q)\times H^{2n-i}(Y,\mathbb Q)\to \mathbb Q$, the isomorphism $H^i_Y(X)\cong H^{i-2c}(Y)$ preserves mixed Hodge structures. This should be compatible with Thom isomorphism. $\endgroup$