Sieved orthogonal polynomials arose when the recurrence coefficients are constant except on an arithmetic progression. Clearly this was too narrow a class. Here an extended class is treated, where the recurrence relation breaks into blocks of the same size, and a particular determinant is constant on these blocks. A symmetric condition assumed earlier in a paper by two of the authors has been dropped, so the new results apply to nonsymmetric sieved polynomials, and also to other interesting cases dealing with polynomial mappings.