  We compute the automorphism group $\mathrm{Aut}(\Delta_{g, n})$ for all $g, n
\geq 0$ such that $3g - 3 + n > 0$, where $\Delta_{g, n} \subset M_{g,
n}^\mathrm{trop}$ is the moduli space of stable $n$-marked tropical curves of
genus $g$ and volume one. In particular, we show that
$\mathrm{Aut}(\Delta_{g})$ is trivial for $g \geq 2$, while
$\mathrm{Aut}(\Delta_{g, n}) \cong S_n$ when $n \geq 1$ and $(g, n) \neq (0,
4), (1, 2)$. The space $\Delta_{g, n}$ is a symmetric $\Delta$-complex in the
sense of Chan, Galatius, and Payne, and is identified with the dual
intersection complex of the boundary divisor in the Deligne-Mumford-Knudsen
moduli space $\overline{\mathcal{M}}_{g, n}$ of stable curves. After the work
of Massarenti, who has shown that $\mathrm{Aut}(\overline{\mathcal{M}}_g)$ is
trivial for $g \geq 2$ while $\mathrm{Aut}(\overline{\mathcal{M}}_{g, n}) \cong
S_n$ when $n \geq 1$ and $2g - 2 + n \geq 3$, our result implies that the
tropical moduli space $\Delta_{g, n}$ faithfully reflects the symmetries of the
algebraic moduli space for general $g$ and $n$.
