We examine a class of multiplicative discrete Painlev{\'{e}} equations which may possess a strongly asymmetric form. When the latter occurs, the equation is written as a system of two equations the right hand sides of which have different functional forms. The present investigation focuses upon two canonical families of the Quispel-Roberts-Thompson classification which contain equations associated with the affine Weyl groups D(1)5 and E(1)6 (or groups appearing lower in the degeneration cascade of these two). Many new discrete Painlev{\'{e}} equations with strongly asymmetric forms are obtained.