  We prove that there are no unexpected universal integral linear relations and
congruences between Hodge, Betti and Chern numbers of compact complex manifolds
and determine the linear combinations of such numbers which are bimeromorphic
or topological invariants. This extends results in the K\"ahler case by
Kotschick and Schreieder. We then define a graded ring which governs the
analogous questions when taking into account 'all' cohomological invariants
(including, for example, the higher pages of the Fr\"olicher spectral sequence,
Bott-Chern and Aeppli cohomology). We partially determine its structure,
allowing for significant generalizations of the initial statements in low
dimensions.
