The quadrupole collectivity of low-lying states and the anomalous behavior of the 02+ and 23+ levels in 72Ge are investigated via projectile multi-step Coulomb excitation with GRETINA and CHICO-2. A total of forty six E2 and M1 matrix elements connecting fourteen low-lying levels were determined using the least-squares search code, gosia. Evidence for triaxiality and shape coexistence, based on the model-independent shape invariants deduced from the Kumar-Cline sum rule, is presented. These are interpreted using a simple two-state mixing model as well as multi-state mixing calculations carried out within the framework of the triaxial rotor model. The results represent a significant milestone towards the understanding of the unusual structure of this nucleus.