The aim of the authors is to give some new characterizations of differential equations of the form \[ \sum_{i=0}^ N l_ i(x) y^{(i)} (x)= \lambda y(x), \tag{*} \] which have polynomial solutions (for certain values of \(\lambda\)) constituting an orthogonal set. Here \(l_ i(x)\) \((i=0,\dots,N)\) are real-valued functions with \(l_ N (x) \not\equiv 0\) and \(\lambda\) is a real parameter. One of the main results of the paper consists of a new proof of a known theorem due to Krall, which makes use of the symmetry equations and is based on the fact that if (*) has orthogonal polynomial solutions, then it is symmetrizable on polynomials. It is also proved that an orthogonal polynomial set satisfies (*) iff it has a certain Sobolev-type orthogonality.