We unravel theoretically a key intrinsic relaxationmechanism among the low-lying singlet and triplet donor-pair states in silicon, an important element in the fast-developing field of spintronics and quantum computation. Despite the perceived weak spin-orbit coupling (SOC) in Si, we find that our discovered relaxation mechanism, combined with the electron-phonon and interdonor interactions, drives the transitions in the two-electron states over a large range of donor coupling regimes. The scaling of the relaxation rate with interdonor exchange interaction J goes from J(5) to J(4) at the low to high temperature limits. Our analytical study draws on the symmetry analysis over combined band, donor envelope, and valley configurations. It uncovers naturally the dependence on the donor-alignment direction and triplet spin orientation, and especially on the dominant SOC source from donor impurities. While a magnetic field is not necessary for this relaxation, unlike in the single-donor spin relaxation, we discuss the crossover behavior with increasing Zeeman energy in order to facilitate comparison with experiments.