  Time-reversal symmetry is a prevalent feature of microscopic physics,
including operational quantum theory and classical general relativity. Previous
works have studied indefinite causal structure using the language of
operational quantum theory, however, these rely on time-asymmetric conditions
to constrain both operations and the process matrix. Here, we use
time-symmetric, operational probabilistic theory to develop a time-symmetric
process matrix formalism for indefinite causal structure. This framework allows
for more processes than previously considered and a larger set of causal
inequalities. We demonstrate that this larger set of causal inequalities offers
new opportunities for device-independent certification of causal
non-separability by violating new inequalities. Additionally, we determined
that the larger class of time-symmetric processes found here is equivalent to
those with Indefinite Causal Order and Time Direction (ICOTD) considered by
Chiribella and Liu, thereby providing a description of these processes in terms
of process matrices.
