The space of Schwarz–Klein spherical triangles

We describe the space of spherical triangles (in the sense of Schwarz and Klein). It is a smooth connected orientable $3$ manifold, homotopy equivalent to the $1$-skeleton of the cubic partition of the closed first octant in $\mathbb{R}^3$. The angles and sides are real analytic functions on this manifold which embed it to $\mathbb{R}^6$.