  The $\mathbb{R}$-motivic cohomology of an $\mathbb{R}$-motivic spectrum is a
module over the $\mathbb{R}$-motivic Steenrod algebra
$\mathcal{A}^{\mathbb{R}}$. In this paper, we describe how to recover the
$\mathbb{R}$-motivic cohomology of the Spanier-Whitehead dual $\mathrm{DX}$ of
an $\mathbb{R}$-motivic finite complex $\mathrm{X}$, as an
$\mathcal{A}^{\mathbb{R}}$-module, given the $\mathcal{A}^{\mathbb{R}}$-module
structure on the cohomology of $\mathrm{X}$. As an application, we show that 16
out of 128 different $\mathcal{A}^{\mathbb{R}}$-module structures on
$\mathcal{A}^{\mathbb{R}}(1):= \langle \mathrm{Sq}^1, \mathrm{Sq}^2 \rangle$
are self-dual.
