  We identify a class of remarkable algebraic relations satisfied by the zeros
of the Krall orthogonal polynomials that are eigenfunctions of linear
differential operators of order higher than two. Given an orthogonal polynomial
family {p_n(x)}, we relate the zeros of the polynomial p_N with the zeros of
p_m for each m <= N (the case m=N corresponding to the relations that involve
the zeros of p_N only). These identities are obtained by exacting the
similarity transformation that relates the spectral and the (interpolatory)
pseudospectral matrix representations of linear differential operators, while
using the zeros of the polynomial p_N as the interpolation nodes. The proposed
framework generalizes known properties of classical orthogonal polynomials to
the case of nonclassical polynomial families of Krall type. We illustrate the
general result by proving new remarkable identities satisfied by the
Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.
