  We prove that every rational extension of the quantum harmonic oscillator
that is exactly solvable by polynomials is monodromy free, and therefore can be
obtained by applying a finite number of state-deleting Darboux transformations
on the harmonic oscillator. Equivalently, every exceptional orthogonal
polynomial system of Hermite type can be obtained by applying a Darboux-Crum
transformation to the classical Hermite polynomials. Exceptional Hermite
polynomial systems only exist for even codimension 2m, and they are indexed by
the partitions \lambda of m. We provide explicit expressions for their
corresponding orthogonality weights and differential operators and a separate
proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3
recurrence relation where l is the length of the partition \lambda. Explicit
expressions for such recurrence relations are given.
