  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
which degrade 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 varies 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
$T_2^*$. 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.
