  Given a family of lattice polytopes, two common questions in Ehrhart Theory
are determining when a polytope has the integer decomposition property and
determining when a polytope is reflexive. While these properties are of
independent interest, the confluence of these properties is a source of active
investigation due to conjectures regarding the unimodality of the
$h^\ast$-polynomial. In this paper, we consider the Newton polytopes arising
from two families of polynomials in algebraic combinatorics: Schur polynomials
and inflated symmetric Grothendieck polynomials. In both cases, we prove that
these polytopes have the integer decomposition property by using the fact that
both families of polynomials have saturated Newton polytope. Furthermore, in
both cases, we provide a complete characterization of when these polytopes are
reflexive. We conclude with some explicit formulas and unimodality implications
of the $h^\ast$-vector in the case of Schur polynomials.
