Silicon quantum dots are considered an excellent platform for spin qubits, partly due to their weak spin-orbit interaction. However, the sharp interfaces in the heterostructures induce a small but significant spin-orbit interaction that degrades the performance of the qubits or, when understood and controlled, could be used as a powerful resource. To understand how to control this interaction, we build a detailed profile of the spin-orbit interaction of a silicon metal-oxide-semiconductor double quantum-dot system. We probe the derivative of the Stark shift, g-factor and g-factor difference for two single-electron quantum-dot qubits as a function of external magnetic field and find that they are dominated by spin-orbit interactions originating from the vector potential, consistent with recent theoretical predictions. Conversely, by populating the double dot with two electrons, we probe the mixing of singlet and spin-polarized triplet states during electron tunneling, which we conclude is dominated by momentum-term spin-orbit interactions that vary from 1.85 MHz up to 27.5 MHz depending on the magnetic field orientation. Finally, we exploit the tunability of the derivative of the Stark shift of one of the dots to reduce its sensitivity to electric noise and observe an 80% increase in T2. We conclude that the tuning of the spin-orbit interaction will be crucial for scalable quantum computing in silicon and that the optimal setting will depend on the exact mode of qubit operations used.