  In the quantum theory, it has been shown that one can see if a process has
the time reversal symmetry by applying the matrix transposition and examining
if it remains physical. However, recent discoveries regarding the indefinite
causal order of quantum processes suggest that there may be other, more general
symmetry transformations of time besides the complete reversal. In this work,
we introduce an expanded concept of matrix transposition, the generalized
transposition, that takes into account general bipartite unitary
transformations of a quantum operation's future and past Hilbert spaces,
allowing for making the time axis definitely lie in a superposed direction,
which generalizes the previously studied `indefinite direction of time', i.e.,
superposition of the forward and the backward time evolution. This framework
may have applications in approaches that treat time and space equally like
quantum gravity, where the spatio-temporal structure is explained to emerge
from quantum mechanics. We apply this generalized transposition to investigate
a continuous generalization of perfect tensors, a dynamic version of tracing
out a subsystem, and the compatibility of multiple time axes in bipartite
quantum interactions. Notably, we demonstrate that when a bipartite interaction
is consistent with more distinct local temporal axes, there is a reduced
allowance for information exchange between the two parties in order to prevent
causality violations.
