Spin-orbit coupling is relatively weak for electrons in bulk silicon, but enhanced interactions are reported in nanostructures such as the quantum dots used for spin qubits. These interactions have been attributed to various dissimilar interface effects, including disorder or broken crystal symmetries. In this Letter, we use a double-quantum-dot qubit to probe these interactions by comparing the spins of separated singlet-triplet electron pairs. We observe both intravalley and intervalley mechanisms, each dominant for [110] and [100] magnetic field orientations, respectively, that are consistent with a broken crystal symmetry model. We also observe a third spin-flip mechanism caused by tunneling between the quantum dots. This improved understanding is important for qubit uniformity, spin control and decoherence, and two-qubit gates.