The structure of the deformed, doubly odd nuclide Re-180 has been studied by gamma-ray and conversion-electron spectroscopy using the Yb-174(B-11,5n) reaction with a pulsed 71 MeV beam of B-11 ions. Several of the previously known intrinsic states have been given revised spin and parity assignments. Rotational bands are observed with K-pi=(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 mu s 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.