On the Steenrod module structure of $\mathbb{R}$-motivic Spanier-Whitehead duals

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 $\mathrm{D}\mathrm{X}$ 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 ${\mathcal{A}}^{\mathbb{R}}(1):=⟨{\mathrm{S}\mathrm{q}}^1,{\mathrm{S}\mathrm{q}}^2⟩$ are self-dual.