  In Ehrhart theory, the $h^*$-vector of a rational polytope often provide
insights into properties of the polytope that may be otherwise obscured. As an
example, the Birkhoff polytope, also known as the polytope of real
doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small
modifications are made to the polytope, the same property can be very difficult
to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes
containing real doubly-stochastic symmetric matrices.
