  In this paper we study combinatorial invariants of the equivalence classes of
pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by
3. These equivalence classes are considered as orbits of lines in
$\mathrm{PG}(3,q)$, under the action of the subgroup $G\cong \mathrm{PGL}(2,q)$
of $\mathrm{PGL}(4,q)$ which preserves the twisted cubic $\mathcal{C}$ in
$\mathrm{PG}(3,q)$. In particular we determine the point orbit distributions
and plane orbit distributions of all $G$-orbits of lines which are contained in
an osculating plane of $\mathcal{C}$, have non-empty intersection with
$\mathcal{C}$, or are imaginary chords or imaginary axes of $\mathcal{C}$.
