Chiral rotation observed in128 Cs is studied using the newly developed microscopic triaxial projected shell model (TPSM) approach. The observed energy levels and the electromagnetic transition probabilities of the nearly degenerate chiral dipole bands in this isotope are well reproduced by the present model. This demonstrates the broad applicability of the TPSM approach, based on a schematic interaction and angular-momentum projection technique, to explain a variety of low-and high-spin phenomena in triaxial rotating nuclei.The classification of band structures from symmetry considerations has played a central role in our understanding of nuclear structure physics. Most of the rotational nuclei are axially symmetric with conserved angular-momentum projection along the symmetry axis. This symmetry has allowed to classify a multitude of rotational bands using Nilsson scheme and has been instrumental to unravel the intrinsic structures of deformed nuclei [1,2]. Although, most of the deformed nuclei obey axial symmetry at low-excitation energies and spin, there are also known regions of the periodic table, referred to as transitional nuclei, that violate axial symmetry and are described using the triaxial deformed mean-field. Further, some nuclei that are axial in the ground-state become triaxial at higher excitation energies and angular momenta. There are several empirical observations indicating that axial symmetry is broken in transitional regions. For instance, the moment of inertia of the transitional nuclei rises sharply with angular-momentum for