  Utilizing dispersive gate sensing (DGS), we investigate the spin-orbit field
($\textbf{B}_{SO}$) orientation in a many-electron double quantum dot (DQD)
defined in an InSb nanowire. While characterizing the inter-dot tunnel
couplings, the measured dispersive signal depends on the electron charge
occupancy, as well as on the amplitude and orientation of the external magnetic
field. The dispersive signal is mostly insensitive to the external field
orientation when a DQD is occupied by a total odd number of electrons. For a
DQD occupied by a total even number of electrons, the dispersive signal is
reduced when the finite external magnetic field aligns with the effective
$\textbf{B}_{SO}$ orientation. This fact enables the identification of
$\textbf{B}_{SO}$ orientations for different DQD electron occupancies. The
$\textbf{B}_{SO}$ orientation varies drastically between charge transitions,
and is generally neither perpendicular to the nanowire nor in the chip plane.
Moreover, $\textbf{B}_{SO}$ is similar for pairs of transitions involving the
same valence orbital, and varies between such pairs. Our work demonstrates the
practicality of DGS in characterizing spin-orbit interactions in quantum dot
systems, without requiring any current flow through the device.
