  A hypercomplex structure $(I,J,K)$ on a manifold $M$ is said to be
$C^\infty$-pure-and-full if the Dolbeault cohomology $H^{2,0}_{\partial}(M,I)$
is the direct sum of two natural subgroups called the $\bar{J}$-invariant and
the $\bar{J}$-anti-invariant subgroups. We prove that a compact hypercomplex
manifold that satisfies the quaternionic version of the $dd^c$-Lemma is
$C^\infty$-pure-and-full. Moreover, we study the dimensions of the
$\bar{J}$-invariant and the $\bar{J}$-anti-invariant subgroups, together with
their analogue in the Bott-Chern cohomology. For instance, in real dimension 8,
we characterize the existence of hyperk\"ahler with torsion metrics in terms of
the dimension of the $\bar{J}$-invariant subgroup. We also study the existence
of special hypercomplex structures on almost abelian solvmanifolds.
