The symmetries of the Hodge diamond of a Kähler manifold provide many non-trivial conditions on both the Hodge numbers and the Betti numbers of the underlying manifold. In this talk I will describe generalizations of these to the setting of Hermitian and almost Kähler manifolds. Both discussions involve zeroeth order terms that vanish in the Kähler case, and yield an interesting representation of sl(2) on a naturally defined subspace of the harmonic forms. I'll explain several topological corollaries involving Betti numbers and the fundamental group, which is joint work with Joana Cirici. The two discussions have some interesting and yet unexplained algebraic mirror-type symmetry between them, which may lead the participants to interesting questions for future research.