  We use the two-pulse spin-dependent photon echo technique to study the
in-plane hole spin anisotropy in a 20~nm-thick CdTe/Cd$_{0.76}$Mg$_{0.24}$Te
single quantum well by exciting the donor-bound exciton resonance. We take
advantage of the photon echo sensitivity to the relative phase of the electron
and hole spin precession and study various interactions contributing to the
hole in-plane spin properties. The main contribution is found to arise from the
crystal cubic symmetry described by the Luttinger parameter $q=0.095$, which is
substantially larger than the one theoretically expected for CdTe or found in
other quantum well structures. Another contribution is induced by the strain
within the quantum well. These two contributions manifest as different
harmonics of the spin precession frequencies in the photon echo experiment,
when strength and orientation of the Voigt magnetic field are varied. The
magnitude of the effective in-plane hole $g$ factor is found to vary in the
range $|\tilde{g_h}|$=0.125--0.160 in the well plane.
