The present paper deals with differential properties of classical orthogonal polynomials in two variables. The authors develop the idea about the connection between polynomials orthogonal with respect to a moment functional in the vector space \(R[x,y]\) of polynomials in two variables with real coefficients, the Pearson condition, and the spectral analysis of certain second order partial differential operators. The classical bivariate orthogonal polynomials come up as the solutions of a matrix second order PDE, which involves the usual gradient and divergence operators. The main result of the paper (Theorem 5) suggests the PDE for successive gradients as a new characterization for bivariate classical orthogonal polynomials. The examples in the last section illustrate the main result.