Block toeplitz products of block toeplitz matrices

Necessary and sufficient conditions for the product of two block Toeplitz matrices to be block Toeplitz are obtained. In the special case of two Toeplitz matrices, the conditions simplify considerably and, when combined with known necessary and sufficient conditions for a nonsingular Toeplitz matrix to have a Toeplitz inverse, provide a simple characterization of the additional matrix structure required by a subclass of Toeplitz matrices in order for it to be closed with respect to both inversion and multiplication.