  In the literature, lines of the projective space $\mathrm{PG}(3,q)$ are
partitioned into classes, each of which is a union of line orbits under the
stabilizer group of the twisted cubic. The least studied class is named
$\mathcal{O}_6$. This class contains lines external to the twisted cubic which
are not its chords or axes and do not lie in any of its osculating planes. For
even and odd $q$, we propose a new family of orbits of $\mathcal{O}_6$ and
investigate in detail their stabilizer groups and the corresponding submatrices
of the point-line and plane-line incidence matrices. To obtain these
submatrices, we explored the number of solutions of cubic and quartic equations
connected with intersections of lines (including the tangents to the twisted
cubic), points, and planes in $\mathrm{PG}(3,q)$.
