  Let $\mathbb{F}$ be a finite field, an algebraically closed field, or the
field of real numbers. Consider the vector space $V=\mathbb{F}^3 \otimes
\mathbb{F}^3$ of $3 \times 3$ matrices over $\mathbb{F}$, and let $G \leq
\text{PGL}(V)$ be the setwise stabiliser of the corresponding Segre variety
$S_{3,3}(\mathbb{F})$ in the projective space $\text{PG}(V)$. The $G$-orbits of
lines in $\text{PG}(V)$ were determined by the first author and Sheekey as part
of their classification of tensors in $\mathbb{F}^2 \otimes V$ in the article
"Canonical forms of $2 \times 3 \times 3$ tensors over the real field,
algebraically closed fields, and finite fields", Linear Algebra Appl. 476
(2015) 133-147. Here we consider the related problem of classifying those line
orbits that may be represented by {\em symmetric} matrices, or equivalently, of
classifying the line orbits in the $\mathbb{F}$-span of the Veronese variety
$\mathcal{V}_3(\mathbb{F}) \subset S_{3,3}(\mathbb{F})$ under the natural
action of $K=\text{PGL}(3,\mathbb{F})$. Interestingly, several of the
$G$-orbits that have symmetric representatives split under the action of $K$,
and in many cases this splitting depends on the characteristic of $\mathbb{F}$.
The corresponding orbit sizes and stabiliser subgroups of $K$ are also
determined in the case where $\mathbb{F}$ is a finite field, and connections
are drawn with old work of Jordan, Dickson and Campbell on the classification
of pencils of conics in $\text{PG}(2,\mathbb{F})$, or equivalently, of pairs of
ternary quadratic forms over $\mathbb{F}$.
