  We introduce and study the crossing map, a closed linear map acting on
operators on the tensor square of a given Hilbert space that is inspired by the
crossing property of quantum field theory. This map turns out to be closely
connected to Tomita--Takesaki modular theory. In particular, crossing symmetric
operators, namely those operators that are mapped to their adjoints by the
crossing map, define endomorphisms of standard subspaces. Conversely, such
endomorphisms can be integrated to crossing symmetric operators. We also
investigate the relation between crossing symmetry and natural compatibility
conditions with respect to unitary representations of certain symmetry groups,
and furthermore introduce a generalized crossing map defined by a real object
in an abstract $C^*$-tensor category, not necessarily consisting of Hilbert
spaces and linear maps. This latter crossing map turns out to be closely
related to the (unshaded, finite-index) subfactor theoretical Fourier
transform. Lastly, we provide families of solutions of the crossing symmetry
equation, solving in addition the categorical Yang--Baxter equation, associated
with an arbitrary Q-system.
