High-spin states in 72Br were studied with the EUROBALL III spectrometer using the 40Ca(40Ca,alpha3p1n) reaction. The negative-parity band observed in this experiment displays a signature inversion around spin I = 16. The interpretation within the cranked Nilsson-Strutinsky approach shows that this signature pattern is a signal of a substantial triaxial shape change with increasing spin where the nucleus evolves from a triaxial shape with rotation about the intermediate axis at low spin through a collective prolate shape to a triaxial shape but with rotation about the shortest principal axis at high spin.