We have measured the spin structure functions g(sub 2)(sup p) and g(sub 2)(sup d) over the kinematic range 0.02(le) x(le) 0.8 and 0.7(le) Q(sup 2)(le) 20 GeV(sup 2) by scattering 29.1 and 32.3 GeV longitudinally polarized electrons from transversely polarized NH(sub 3) and (sup 6)LiD targets. Our measured g(sub 2) approximately follows the twist-2 Wandzura-Wilczek calculation. The twist-3 reduced matrix elements d(sub 2)(sup p) and d(sub 2)(sup n) are less than two standard deviations from zero. The data are inconsistent with the Burkhardt-Cottingham sum rule. The Efremov-Leader-Teryaev integral is consistent with zero within our measured kinematic range.