An extended decay scheme for Xe-128 has been constructed by using data from the Sn-124(Be-9, 5n)Xe-128 reaction at a beam energy of 58 MeV. Bands have been identified as being built on several intrinsic states, including a proposed 9/2(-)[514] circle times 1/2(+)[400] two-quasineutron configuration that forms the K-pi=5(-) intrinsic state at 2228 keV, and on a previously assigned K-pi=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 K-pi=5(-) intrinsic state, whereas a nearly axially symmetric shape is predicted for the K-pi=8(-) two-quasiparticle configuration. The low value of the hindrance factor for the E1 transition depopulating the K-pi=8(-) intrinsic state is discussed in the context of analogous transitions in neighbouring N=74 isotones.