Abstract: The beta-delayed gamma-ray spectroscopy of neutron-rich Rh-117,Rh-119 isotopes has been investigated at the Radioactive Isotope Beam Factory of RIKEN. The 1/2(-) isomers in Rh-117,Rh-119 have been identified for the first time. The systematics of energy differences between the 9/2(+) and 1/2(-) states have been extended up to N = 74, which shows a tendency to first increase with the neutron number N, then reach a maximum at N approximate to 68, and subsequently decrease. Self-consistent triaxial relativistic Hartree-Bogoliubov (TRHB) calculations highlight the crucial role of triaxial deformation, particularly in the positive-parity states, in driving the observed evolution of energy differences. A shape transition from axially symmetric to triaxial and back to axially symmetric deformation is suggested along the Rh isotopic chain. These findings provide new insights into the structure of neutron-rich nuclei in the A similar to 110 region and motivate further studies of exotic isotopes in this region to explore the interplay between triaxiality and shell evolution.