We present a time-reversible Born-Oppenheimer molecular dynamics scheme, based on self-consistent Hartree-Fock or density functional theory, where both the nuclear and the electronic degrees of freedom are propagated in time. We show how a time-reversible adiabatic propagation of the electronic degrees of freedom is possible despite the nonlinearity and incompleteness of the self-consistent field procedure. With a time-reversible lossless propagation the simulated dynamics is stabilized with respect to a systematic long-term energy drift and the number of self-consistency cycles can be kept low thanks to a good initial guess given from the electronic propagation. The proposed molecular dynamics scheme therefore combines a low computational cost with a physically correct time-reversible representation, which preserves a detailed balance between propagation forwards and backwards in time.