We measured the spin structure functions g₂^p and g₂^d in the range 0.02\leq x \leq 0.8 and 0.7 \leq Q² \leq 20 GeV² by scattering 29.1 and 32.3 GeV longitudinally polarized electrons from transversely polarized NH₃ and ⁶{LiD} targets. g₂ approximately follows the twist-2 Wandzura--Wilczek calculation. The twist-3 reduced matrix elements d₂^p and d₂ⁿ are less than two standard deviations from zero. The data are inconsistent with the Burkhardt--Cottingham sum rule if there is no pathological behavior as x \rightarrow 0. The Efremov--Leader--Teryaev integral is consistent with zero.

PACS numbers: 13.60.Hb, 13.88.+e, 24.70.+s, 25.30.Fj