  The BNSR-invariants of a group $G$ are a sequence $\Sigma^1(G)\supseteq
\Sigma^2(G) \supseteq \cdots$ of geometric invariants that reveal important
information about finiteness properties of certain subgroups of $G$. We
consider the symmetric automorphism group $\Sigma Aut_n$ and pure symmetric
automorphism group $P\Sigma Aut_n$ of the free group $F_n$, and inspect their
BNSR-invariants. We prove that for $n\ge 2$, all the ``positive'' and
``negative'' character classes of $P\Sigma Aut_n$ lie in $\Sigma^{n-2}(P\Sigma
Aut_n)\setminus \Sigma^{n-1}(P\Sigma Aut_n)$. We use this to prove that for
$n\ge 2$, $\Sigma^{n-2}(\Sigma Aut_n)$ equals the full character sphere $S^0$
of $\Sigma Aut_n$ but $\Sigma^{n-1}(\Sigma Aut_n)$ is empty, so in particular
the commutator subgroup $\Sigma Aut_n'$ is of type $F_{n-2}$ but not $F_{n-1}$.
Our techniques involve applying Morse theory to the complex of symmetric marked
cactus graphs.
