We use tools from the theory of dynamical systems with symmetries to stratify Uhlmann's standard purification bundle and derive a new connection for mixed quantum states. For unitarily evolving systems, this connection gives rise to the 'interferometric' geometric phase of Sjqvist et al (2000 Phys. Rev. Lett. 85 2845-9), and for more generally evolving open systems it gives rise to the generalization of the interferometric geometric phase due to Tong et al (2004 Phys. Rev. Lett. 93 080405).