  We present a new, geometric perspective on the recently proposed triality of
2d $\mathcal{N}=(0,1)$ gauge theories, based on its engineering in terms of
D1-branes probing Spin(7) orientifolds. In this context, triality translates
into the fact that multiple gauge theories correspond to the same underlying
orientifold. We show how Spin(7) orientifolds based on a particular involution,
which we call the universal involution, give rise to precisely the original
version of $\mathcal{N}=(0,1)$ triality. Interestingly, our work also shows
that the space of possibilities is significantly richer. Indeed, general
Spin(7) orientifolds extend triality to theories that can be regarded as
consisting of coupled $\mathcal{N}=(0,2)$ and $(0,1)$ sectors. The geometric
construction of 2d gauge theories in terms of D1-branes at singularities
therefore leads to extensions of triality that interpolate between the pure
$\mathcal{N}=(0,2)$ and $(0,1)$ cases.
