We study the orthogonal polynomials of classical and semi-classical types that can be defined by ordinary and exponential Riordan arrays. We identify their moment sequences, giving their integral representations and Hankel transforms. For a special class of classical orthogonal polynomials defined by Riordan arrays, we identify a complementary family of orthogonal polynomials defined by reversion of moment sequences. Special product sequences arise and their generating functions are calculated.