  We consider the four-component two-valley Dirac operator on a wedge in
$\mathbb{R}^2$ with infinite mass boundary conditions, which enjoy a flip at
the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize
all its self-adjoint extensions, relying on the fact that the underlying
two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The
respective defect element is computed explicitly.
  We observe that there exists no self-adjoint extension, which can be
decomposed into an orthogonal sum of two two-component operators. In physics,
this effect is called mixing the valleys.
