  Using a generalized RPA-type theory we calculate the in-plane anisotropy of
the magnetic excitations in hole-doped high-$T_c$ superconductors. Extending
our earlier Fermi-liquid based studies on the resonance peak by inclusion of
orthorhombicity we still find two-dimensional spin excitations, however, being
strongly anisotropic. This reflects the underlying anisotropy of the hopping
matrix elements and of the resultant superconducting gap function. We compare
our calculations with new experimental data on {\it fully untwinned}
${YBa}_2{Cu}_3{O}_{6.85}$ and find good agreement. Our results are in contrast
to earlier interpretations on the in-plane anisotropy in terms of stripes (H.
Mook {\it et al.}, Nature {\bf 404}, 729 (2000)), but reveal a conventional
solution to this important problem.
