  It was recently conjectured that every system of exceptional orthogonal
polynomials is related to classical orthogonal polynomials by a sequence of
Darboux transformations. In this paper we prove this conjecture, which paves
the road to a complete classification of all exceptional orthogonal
polynomials. In some sense, this paper can be regarded as the extension of
Bochner's result for classical orthogonal polynomials to the exceptional class.
As a supplementary result, we derive a canonical form for exceptional operators
based on a bilinear formalism, and prove that every exceptional operator has
trivial monodromy at all primary poles.
