A class of orthogonal polynomials relative to special discrete weights is considered. These self-dual weights which are completely determined by their finite support appear in the polynomial approximation of a function over this set, in the barycentric form of classical Lagrange interpolation polynomial as well as in mathematical statistics and statistical physics. Orthogonal polynomials possess remarkable symmetry reflected in the properties of their coefficients in the three-term recurrence relations, some of which have an explicit form. Formulas for the expected values are derived. •The orthogonal polynomials with regard to self-dual weights are known to satisfy a special three-term recurrence with a persymmetric Jacobi matrix, and this is a characteristic property of self-dual weights;•The corresponding moments satisfy the recurrence which translates into factorization formula for associate polynomials;•Explicit form of central orthogonal polynomials allows for asymptotic study when the weights are random. This study indicates importance of the maximal attraction domain of the underlying distribution;•A form of Central Limit Theorem for the coefficients of studied polynomials is suggested.