A detailed inelastic neutron scattering study of the in-plane anisotropy of magnetic excitations in twin-free YBa\(_2\)Cu\(_3\)O\(_{6.6}\) (\(T_c =\) 61 K) reveals that the spin excitation spectra in the superconducting and normal states are qualitatively different. Below \(T_c\), the spectrum consists of upward- and downward-dispersing branches with modest in-plane anisotropy merging at an energy \(E_{res} =\) 37.5 meV. In the normal state, the singularity at \(E_{res}\) disappears, and the spectrum exhibits a steep dispersion with a strongly anisotropic in-plane geometry. These data have important implications for models based on static or dynamic "stripe" order of spins and charges.