Let $V$ be a two-dimensional real vector space and let $s, t$ be two linear reflections on $V$ with the same $(+1)$-eigenspace. I would like to show that the dual maps $s^*$ and $t^*$ in the dual space $V^*$ have the same $(-1)$-eigenspace.
A linear reflection $T : V \to V$ is a map for which there are $H, H'$ $1$-dimensional subspaces of $V$ with $$V = H \oplus H', $$ and $$T|_{H} = \mathrm{Id}, T|_{H'} = -\mathrm{Id}, $$ i.e. a linear reflection is a generalization of an orthogonal reflection.
I have been able to show that two linear reflections have the same $(-1)$-eigenspace if and only if there is some affine line not passing through the origin which is invariant under the action of both $s$ and $t$ (this is because, such affine lines must be parallel to the $(-1)$-eigenspace).
It is clear that the dual maps $s^*$ and $t^*$ are indeed linear reflections on $V^*$ (as if $v_1, v_2$ is a basis for $V$ consisting of eigenvectors of $s$, the dual basis of $v_1, v_2$ will make $s^*$ a linear reflection, and likewise for $t^*$). So, using the above result, we only need to show that $s^*$ and $t^*$ have a common in variant affine line not passing through the origin, but I have not been able to prove this.