The names para- and ortho-exciton stem from the fact that -- in superficial analogy to para- and ortho-hydrogen -- the wave function that forms the electron-hole bound state can either be a singlet state with vanishing total spin or a triplet state with total spin $1$. In a semiconductor that shows strong spin-orbit interaction $S^2$ and $S_z$ are no good quantum numbers any more. Therefore the para- and ortho-exciton states can not be eigenstates of these operators.
Is it possible to still make this distinction for the exciton in the case of spin-orbit coupling? If not, is there a redefinition of how to distinguish para- and ortho-excitons?
