I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be grateful if the experts would point out any subtleties with my questions if rational is replaced by real. Let me at the moment focus on questions related to polarizability:

Let $\mathcal{L}$ be a local system underlying a polarizable variation of Hodge structure on a smooth variety. Does the polarizability imply $\mathcal{L}$ is self-dual? If yes, then does every direct summand of this local system also have to be self-dual (I am guessing no to the latter)?

More generally, let $M$ be a polarizable mixed Hodge module on some variety (not necessarily smooth). Is polarizability equivalent to Verdier self-duality (up to Tate twist) in the derived category of mixed Hodge modules? If not equivalent, does it at least imply Verdier self-duality?