  What does it mean for a causal structure to be `unknown'? Can we even talk
about `repetitions' of an experiment without prior knowledge of causal
relations? And under what conditions can we say that a set of processes with
arbitrary, possibly indefinite, causal structure are independent and
identically distributed? Similar questions for classical probabilities, quantum
states, and quantum channels are beautifully answered by so-called "de Finetti
theorems", which connect a simple and easy-to-justify condition -- symmetry
under exchange -- with a very particular multipartite structure: a mixture of
identical states/channels. Here we extend the result to processes with
arbitrary causal structure, including indefinite causal order and multi-time,
non-Markovian processes applicable to noisy quantum devices. The result also
implies a new class of de Finetti theorems for quantum states subject to a
large class of linear constraints, which can be of independent interest.
