The band structures of the doubly odd 1 3 4 Pr nucleus has been investigated via the 1 1 9 Sn( 1 9 F,4n) 1 3 4 Pr and 1 1 0 Pd( 2 8 Si,p3n) 1 3 4 Pr reactions at beam energies of 87 and 130 MeV, respectively. The three previously known rotational bands based on the π[541]3 - (α=±1) ν[514]9 - and π[413]5 + ν[514]9 - configurations have been extended to higher spins. The difference of 2 in the experimental alignment of the bands based on the signature partners of the [541]3 - h 1 1 proton orbital is discussed in terms of shape coexistence and coupling with the γ-phonon but no consistent interpretation can be found. A new band consisting of quadrupole transitions has also been identified and linked unambiguously to the low-lying levels of the yrast band. Total routhian surface and cranked shell model calculations suggest a (πh 1 1 ) 3 ν[530]1 - configuration for this band. The experimental ratios of reduced transition probabilities B(M1)/B(E2) of the various rotational structures are compared to the theoretical values obtained from a semiclassical model and from the two-particle plus triaxial-rotor model.