Spectrahedra are linear sections of the cone of positive semidefinite matrices which, as convex bodies, generalize the class of polyhedra. In this paper we investigate the problem of recognizing when a spectrahedron is polyhedral. We generalize and strengthen results of [M. V. Ramana, Polyhedra, spectrahedra, and semidefinite programming, in Topics in Semidefinite and Interior-Point Methods, Fields Inst. Commun. 18, AMS, Providence, RI, 1998, pp. 27-38] regarding the structure of spectrahedra, and we devise a normal form of representations of spectrahedra. This normal form is effectively computable and leads to an algorithm for deciding polyhedrality.