  In this paper, we study irreducible non-weight modules over the mirror
Heisenberg-Virasoro algebra $\mathcal{D}$, including Whittaker modules,
$\mathcal{U}(\mathbb{C} d_0)$-free modules, and their tensor products. More
precisely, we give the necessary and sufficient conditions for the Whittaker
modules to be irreducible. We determine all $\mathcal{D}$-module structures on
$\mathcal{U}(\mathbb{C} d_0)$, and find the necessary and sufficient conditions
for these modules to be irreducible. At last we determine the necessary and
sufficient conditions for the tensor products of Whittaker modules and
$\mathcal{U}(\mathbb{C} d_0)$-free modules to be irreducible, and obtain that
any two such tensor products are isomorphic if and only if the corresponding
Whittaker modules and $\mathcal{U}(\mathbb{C} d_0)$-free modules are
isomorphic. These lead to many new irreducible non-weight modules over
$\mathcal{D}$.
