  We investigate a new case of rigidity in stable homotopy theory which is the
rigidity of the $K(1)$-local stable homotopy category
$\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ at $p=2$. In other words, we show that
recovering higher homotopy information by just looking at the triangulated
structure of $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ is possible, which is a
property that only few interesting stable model categories are known to
possess.
