Three new negative-parity bands have been identified in 120Ba, two of them forming a strongly coupled band. The previously known negative-parity band is significantly extended to high spin, while the lower part of the yrare positive-parity band has been modified. From the analysis of the band properties and comparison with the neighboring nuclei a coherent description of all bands is achieved. In particular, a simple explanation of the evolution of the positive-parity bands at high spin is proposed, including the possible occupation of the ??f7/2[541]1/2??? intruder orbital. Cranked Nilsson-Strutinsky calculations reveal similar quadrupole deformations but different triaxiality of the bands, while particle number conserving cranked shell model calculations qualitatively reproduce the experimental data and support the assigned configurations. The new measured ratios of reduced transition probabilities B(E1)/B(E2) complete the systematics in the 118???124Ba nuclei, exhibiting a decrease with decreasing neutron number, and are compared with the known values in the 116???120Xe nuclei, which are larger. Extended calculations with the quadrupole and octupole collective Hamiltonian based on the relativistic Hartree-Bogoliubov model employing the relativistic DD-PC1 density functional nicely reproduce the decreasing trend towards lower neutron numbers for Ba and Xe nuclei, as well as the larger values in Xe nuclei, but are much larger in amplitude than the experimental values. On the other hand, particle number conserving cranked shell model calculations without octupole deformation overestimate the low-spin values, while those with octupole deformation included reproduce the experimental values in 120Ba, suggesting the possible existence of moderate octupole collectivity in the negative-parity bands of nuclei in this mass region.