We report the measurement of two new perpendicular (D2O)(3) torsional bands by terahertz laser vibration-rotation-tunneling (VRT) spectroscopy of a planar pulsed supersonic expansion. The first (28.0 cm(-1)) band corresponds to the k = +/- 2(l) <-- 0 transition, and is the lowest frequency vibrational spectrum observed for a water cluster. The second (81.8 cm(-1)) band originates in the first excited torsional state, and has been assigned as k = 3u <-- 1(l). An effective three- dimensional Hamiltonian is derived to describe the rotational structure of each torsional state. Degenerate torsional levels with k = +/-1 and k = +/-2 exhibit a Coriolis splitting linear in K implying the presence of vibrational angular momentum, and a second-order splitting from off- diagonal coupling between degenerate sublevels with + \k\ and - \k\. With this effective Hamiltonian we fit a total of 554 rovibrational transitions in five different bands connecting the lowest nine torsional states, with a rms residual of 1.36 MHz. The data set comprises the two new VRT bands together with the 41.1 cm(-1) parallel band, the 89.6 cm(-1) parallel band, and the 98.1 cm(-1) perpendicular band. This analysis provides a comprehensive characterization of the torsional energy levels in (D2O)(3) up to 100 cm(-1) above the zero- point energy, and confirms the torsional assignments for all five (D2O)(3) VRT bands observed to date. Moreover, it vindicates the adiabatic separation of the trimer torsional and hydrogen bond stretch/bend vibrational modes which underlies the torsional model.