We present the measurement and analysis of five new (D2O)5 bands via vibration-rotation-tunneling (VRT) spectroscopy as well as a preliminary description of a second (H2O)5 band. The vibrationally averaged rotational constants of all five fitted bands agree well with those from the two previously observed (D2O)5 bands and confirm that the pentamer averages to a symmetric, quasi-planar structure on the time scale of our experiment. While the spectrum of the first two bands, located at 50.7 cm(-1) (1.52 THz) and 27.3 cm(-1) (0.82 THz) are indicative of unperturbed oblate rotors, the three remaining (D2O)5 bands centered at 47.7 cm(-1) (1.43 THz), 45.4 cm(-1) (1.36 THz), and 45.0 cm(-1) (1.35 THz) are severely perturbed by first-order Coriolis coupling. This represents the first observation of this perturbation in the perdeuterated water pentamer, as well as the first observation of transitions between degenerate states of the torsional-puckering manifold. Unlike transitions from the (H2O)5 band observed by Brown et al. at 89.0 cm(-1) and the 103.8 cm(-1) band that we report here, none of the individual rovibrational transitions of any of the five (D2O)5 bands demonstrate spectral splittings due to bifurcation tunneling. We conclude, through careful analysis of these water pentamer bands, that at least three torsional-puckering manifolds have been probed and that the lowest-energy manifold is highly compacted. A plausible water pentamer torsional-puckering correlation diagram is proposed, though additional experimental data are required to unambiguously establish the energies of the torsional-puckering levels.