High-temperature superconductors are nowadays found in great variety and hold technological promise. It is still an unsolved mystery that the critical temperature T of the basic cuprates is so high. The answer might well be hidden in a conventional corner of theoretical physics, overlooked in the recent hunt for exotic explanations of new effects in these materials. A forgotten intra-atomic s-d two-electron exchange in the Cu atom is found to provide a strong (similar to eV) electron pairing interaction. A Bardeen-Cooper-Schrieffer approach can explain the main experimental observations and predict the correct d(x2-y2) symmetry of the gap.