  Mixed-norm $\alpha$-modulation spaces were introduced recently by Cleanthous
and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. The
mixed-norm spaces $M^{s,\alpha}_{\vec{p},q}(\mathbb{R}^n)$, $\alpha\in [0,1]$,
form a family of smoothness spaces that contain the mixed-norm Besov spaces as
special cases. In this paper we prove that a pseudodifferential operator
$\sigma(x,D)$ with symbol in the H\"ormander class $S^b_{\rho}$ extends to a
bounded operator $\sigma(x,D)\colon M^{s,\alpha}_{\vec{p},q}(\mathbb{R}^n)
\rightarrow M^{s-b,\alpha}_{\vec{p},q}(\mathbb{R}^n)$ provided $0<\alpha\leq
\rho\leq 1$,
  $\vec{p}\in (0,\infty)^n$, and $0<q<\infty$. The result extends the known
result that pseudodifferential operators with symbol in the class $S^b_{1}$
maps the mixed-norm Besov space $B^s_{\vec{p},q}(\mathbb{R}^n)$ into
$B^{s-b}_{\vec{p},q}(\mathbb{R}^n)$.
