Inelastic neutron scattering has been used to obtain a comprehensive description of the absolute dynamical spin susceptibility chi ''(q, omega) of the underdoped superconducting cuprate YBa2Cu3O6.5 (T-c = 52 K) over a wide range of energies and temperatures (2 meV less than or equal to (h) over bar omega less than or equal to 120 meV and 5 K less than or equal to T less than or equal to 200 K). Spin excitations of two distinct symmetries (even and odd under the exchange of two adjacent CuO2 layers) are observed which exhibit two different gaplike features (rather than a single ''spin pseudogap''). The excitations show dispersive behavior at high energies. [S0163-1829(97)51142-8].