The structure of the deformed, doubly odd nuclide 180Re has been studied by gamma-ray and conversion-electron spectroscopy using the 174Yb(11B,5n) reaction with a pulsed 71 MeV beam of 11B ions. Several of the previously known intrinsic states have been given revised spin and parity assignments. Rotational bands are observed with Kpi=(4+),(5-),(7+),8+,9-,13+,14-,15-,16+,21-, and (22+). Among these, a four-quasiparticle t band is identified, which is already energetically favored at its bandhead compared to the corresponding two-quasiparticle band; and two six-quasiparticle bands are identified and associated with a tau=13 µs isomer. The observed structures, including g factors and alignments, are interpreted with the aid of Nilsson-plus-BCS calculations and configuration-constrained potential energy surface calculations. Reduced-hindrance values are obtained for K-forbidden transitions, illustrating the important role of the K quantum number for near-yrast isomers.