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It was recently found that it is theoretically possible for there to exist higher-order quantum processes in which the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes are believed to have a physical realization in standard quantum mechanics via coherent control of the times of the operations. A prominent example is the quantum SWITCH, which was recently demonstrated experimentally. However, up until now, there has been no rigorous justification for the interpretation of such an experiment as a genuine realization of a process with indefinite causal structure as opposed to a simulation of such a process. Where exactly are the local operations of the parties in such an experiment? On what spaces do they act given that their times are indefinite? Can we probe them directly rather than assume what they ought to be based on heuristic considerations? How can we reconcile the claim that these operations really take place, each once as required, with the fact that the structure of the presumed process implies that they cannot be part of any acyclic circuit? Here, I offer a precise answer to these questions: the input and output systems of the operations in such a process are generally nontrivial subsystems of Hilbert spaces that are tensor products of Hilbert spaces associated with different times—a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is one of a circuit with a cycle, which cannot be reduced to a (possibly dynamical) probabilistic mixture of acyclic circuits. This provides, for the first time, a rigorous proof of the existence of processes with indefinite causal structure in quantum mechanics. I further show that all bipartite processes that obey a recently proposed unitary extension postulate, together with their unitary extensions, have a physical realization on such time-delocalized subsystems, and provide evidence that even more general processes may be physically admissible. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.