  There are natural polynomial invariants of polytopes and lattice polytopes
coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and
$h^*$-polynomials, respectively. In this paper, we study their generalization
to subdivisions and lattice subdivisions of polytopes. By abstracting
constructions in mixed Hodge theory, we introduce multivariable polynomials
which specialize to the $h$-, $h^*$- polynomials. These polynomials, the mixed
$h$-polynomial and the (refined) limit mixed $h^*$-polynomial have rich
symmetry, non-negativity, and unimodality properties, which both refine known
properties of the classical polynomials, and reveal new structure. For example,
we prove a lower bound theorem for a related invariant called the local
$h^*$-polynomial. We introduce our polynomials by developing a very general
formalism for studying subdivisions of Eulerian posets that extends the work of
Stanley, Brenti and Athanasiadis on local $h$-vectors. In particular, we prove
a conjecture of Nill and Schepers, and answer a question of Athanasiadis.
