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We report a comprehensive polarized and unpolarized neutron scattering study of the evolution of the dynamical spin susceptibility with temperature and doping in three underdoped single crystals of the YBa2Cu3O6+x high temperature superconductor: YBa2Cu3O6.5 (Tc = 52 K), YBa2Cu3O6.7 (Tc = 67 K), and YBa2Cu3O6.85 (Tc = 87 K). The spin susceptibility is determined in absolute units at excitation energies between 1 and 140 meV and temperatures between 1.5 and 300 K. Polarization analysis is used extensively at low energies. Transitional matrix elements, including those between spin states, in a bilayer system such as YBa2Cu3O6+x can be generally classified into even and odd, according to the sign change under a symmetry operation that exchanges the layers, and both even and odd excitations are detected in YBa2Cu3O6.5 and YBa2Cu3O6.7. While the even spin excitations show a true gap which depends on doping, the odd spectrum is characterized by a weakly dopingdependent pseudogap. Both even and odd components are substantially enhanced upon lowering the temperature from 300 K. The even excitations evolve smoothly through the superconducting transition temperature Tc, but the odd excitations develop a true gap below Tc. At the same time, the odd spin susceptibility is sharply enhanced below Tc around an energy that increases with doping. This anomaly in the magnetic spectrum is closely related to the magnetic resonance peak that appears at 40 meV in the superconducting state of the optimally doped compound (Tc = 93 K). From these data we extract the energy and the energy-integrated spectral weight of the resonance peak in absolute units as a function of doping level. Theoretical implications of these measurements are discussed, and a critique of recent attempts to relate the spin excitations to the thermodynamics of high temperature superconductors is given.