  The states of two electrons in tunnel-coupled semiconductor quantum dots can
be effectively described in terms of a two-spin Hamiltonian with an isotropic
Heisenberg interaction. A similar description needs to be generalized in the
case of holes due to their multiband character and spin-orbit coupling, which
mixes orbital and spin degrees of freedom, and splits $J=3/2$ and $J = 1/2$
multiplets. Here we investigate two-hole states in prototypical coupled Si and
Ge quantum dots via different theoretical approaches. Multiband
$\boldsymbol{k}\cdot\boldsymbol{p}$ and Configuration-Interaction calculations
are combined with entanglement measures in order to thoroughly characterize the
two-hole states in terms of band mixing and justify the introduction of an
effective spin representation, which we analytically derive a from generalized
Hubbard model. We find that, in the weak interdot regime, the ground state and
first excited multiplet of the two-hole system display -- unlike their
electronic counterparts -- a high degree of $J$-mixing, even in the limit of
purely heavy-hole states. The light-hole component additionally induces
$M$-mixing and a weak coupling between spinors characterized by different
permutational symmetries.
