The high-spin decay of ${}^{71}$Se was studied using the ${}^{54}$Fe${(}^{23}$Na,$\ensuremath{\alpha}pn)$ reaction at 80 MeV and the Florida State University Compton-suppressed Ge array consisting of three clover detectors and seven single-crystal detectors. Based on prompt $\ensuremath{\gamma}$-$\ensuremath{\gamma}$ coincidences measured in the experiment, the known level scheme was enhanced and extended to higher spin with 19 new transitions. A band that was previously suggested to have positive parity was reassigned as the ``missing'' signature partner of an existing negative-parity band. Spins were assigned based on directional correlation of oriented nuclei ratios. Lifetimes of 17 excited states were measured using the Doppler-shift attenuation method. Experimental ${Q}_{t}$ values imply an intermediate degree of collective behavior for ${}^{71}$Se at high spin. Theoretical ${Q}_{t}$ values determined from cranked Woods-Saxon (CWS) calculations show better agreement with the experimental ones for the positive-parity states than the negative-parity states. Shape competition and $\ensuremath{\gamma}$ softness characterize the low-spin states of the lowest positive- and negative-parity bands based on the CWS calculations. At high spin, triaxial shapes with $\ensuremath{\gamma}g{0}^{\ensuremath{\circ}}$ are predicted.