Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable invariants of subdivisions of polytopes, lattice polytopes and fans, namely the mixed $h$-polynomial, the refined limit mixed $h^*$-polynomial, and the mixed $cd$-index, defined by Katz--Stapledon and Dornian--Katz--Tsang. Previously, only the refined limit mixed $h^*$-polynomial had a geometric interpretation, which came from filtrations on the cohomology of a schön hypersurface. Consequently, we generalize a positivity result on the mixed $h$-polynomial by Katz and Stapledon using the relative hard Lefschetz theorem of Karu.