  We determine the invariants characterizing the $Sp(n)$-orbits in the real
Grassmannian $Gr^\R(2k,4n)$ of the $2k$-dimensional complex and
$\Sigma$-complex subspaces of a $4n$-dimensional Hermitian quaternionic vector
space. A $\Sigma$-complex subspace is the orthogonal sum of complex subspaces
by different, up to sign, compatible complex structure. The result is obtained
by considering two main features of such subspaces. The first is that any such
subspace admits a decomposition into an Hermitian orthogonal sum of
4-dimensional complex addends plus a 2-dimensional totally complex subspace if
$k$ is odd, meaning that the quaternionification of the addends are orthogonal
in pairs. The second is that any 4-dimensional complex addend $U$ is an
isoclinic subspace i.e. the principal angles of the pair $(U,AU)$ are all the
same for any compatible complex structure $A$. Using these properties we
determine the full set of the invariants characterizing the $Sp(n)$-orbit of
such subspaces in $Gr^\R(2k,4n)$.
