We detect a radiative cascade which initiates from a metastable biexciton state in a neutral semiconductor quantum dot. In this biexciton, the heavy holes form a spin-triplet configuration, Pauli blockaded from relaxation to the spin-singlet ground state. The triplet biexciton has two photon-phonon-photon decay paths. Unlike in the singlet-ground-state biexciton radiative cascade, in which the two photons are colinearly polarized, in the triplet-biexciton cascade they are cross-linearly polarized. We measured the two-photon polarization density matrix and show that the phonon emitted when the intermediate exciton relaxes from excited to ground state, preserves the exciton’s spin. The phonon, thus, does not carry with it any which-path information other than its energy. Nevertheless, entanglement distillation by spectral filtering was found to be rather ineffective for this cascade. This deficiency results from the opposite sign of the anisotropic electron-hole exchange interaction in the excited exciton relative to that in the ground exciton.