An extended decay scheme for 128Xe has been constructed by using data from the 124Sn(9Be, 5n)128Xe reaction at a beam energy of 58 MeV. Bands have been identified as being built on several intrinsic states, including a proposed 9/2- [direct-product]1/2+ two-quasineutron configuration that forms the Kpi=5- intrinsic state at 2228 keV, and on a previously assigned Kpi=8- intrinsic state at 2786 keV. A half-life of 73(3) ns has been measured for the latter. Theoretical calculations have been performed by using the configuration-constrained blocking method based on a nonaxial Woods-Saxon potential. Large gamma deformation and gamma softness are predicted for the ground state and the Kpi=5- intrinsic state, whereas a nearly axially symmetric shape is predicted for the Kpi=8- two-quasiparticle configuration. The low value of the hindrance factor for the E1 transition depopulating the Kpi=8- intrinsic state is discussed in the context of analogous transitions in neighbouring N=74 isotones.