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Given a finite set $F=\{f_1,\cdots ,f_k\}$ of nonnegative integers (writtenin increasing size) and a classical discrete family $(p_n)_n$ of orthogonalpolynomials (Charlier, Meixner, Krawtchouk or Hahn), we consider the Casoratideterminant $\det(p_{f_i}(x+j-1))_{i,j=1,\cdots,k}$. In this paper we prove anice invariant property for this kind of Casorati determinants when the set $F$is changed by $I(F)=\{0,1,2,\cdots,\max F\}\setminus \{\max F-f:f\in F\}$. Thissymmetry is related to the existence of higher order difference equations forthe orthogonal polynomials with respect to certain Christoffel transforms ofthe classical discrete measures. By passing to the limit, this invariantproperty is extended for Wronskian type determinants whose entries are Hermite,Laguerre and Jacobi polynomials.

Invariant properties for Wronskian type determinants of classical and  classical discrete orthogonal polynomials under an involution of sets of  positive integers