Doubly Cohen-Macaulay complexes are a family of simplicial complexes that includes triangulations of spheres, Bergman complexes of matroids, and independence complexes of coloopless matroids. Work of Adiprasito, Papadakis, and Petrotou implies that the $h$-vector of a doubly Cohen-Macaulay complex of dimension $d - 1$ satisfies $h_i \le h_{d-i}$ for $i \le d/2$. The complementary vector is $(h_d - h_0, h_{d-1} - h_1, \dotsc)$. We show that the complementary vector of a doubly Cohen-Macaulay complex is the Hilbert function of a module over a polynomial ring which is generated in degree 0, and that any such Hilbert function is the complementary vector of a doubly Cohen-Macaulay complex. Joint work with Alan Stapledon.