Following two previous quantum dynamics studies [J. Chem. Phys. 97, 6784 (1992); 99, 1373 (1993)], we present in this paper a more thorough investigation of the symmetry and rotational orientation effects in dissociative chemisorption of diatomic molecules on metals. Specifically, we extended our theoretical studies to calculate the sticking coefficients for H, and its isotopomer HD on Cu from all angular momentum states (up to j = 8). Our calculation shows a strong dependence of the dissociation probability P(jm) on both j and m rotation quantum numbers, and the increases of P(jm) are closely correlated with the increase of the quantum number m in a given j manifold. Also the dissociation of the diatomic rotational states whose quantum numbers satisfy j+m = odd is forbidden at low energies for the homonuclear H-2 due to the selection rule. The present study provides further evidence that the effect of diatomic rotation on adsorption mainly arises from the effect of rotational orientation (m dependence) as found in previous studies. This m dependence predicts that at low kinetic energies, the degeneracy-averaged dissociation probability of hydrogen on Cu increases monotonically as the rotation quantum number j increases. However, at high kinetic energies, the adsorption probability first decreases as j increases from 0 to about 4-5 before increasing as j further increases above 4-5. The latter behavior is consistent with a recent experimental measurement by Michelsen et al. of the mean kinetic energy of the rotational states of D-2 desorbed from Cu(111).