  The symmetry of quantum theory under time reversal has long been a subject of
controversy because the transition probabilities given by Born's rule do not
apply backward in time. Here, we resolve this problem within a rigorous
operational probabilistic framework. We argue that reconciling time reversal
with the probabilistic rules of the theory requires a notion of operation that
permits realizations via both pre- and post-selection. We develop the
generalized formulation of quantum theory that stems from this approach and
give a precise definition of time-reversal symmetry, emphasizing a previously
overlooked distinction between states and effects. We prove an analogue of
Wigner's theorem, which characterizes all allowed symmetry transformations in
this operationally time-symmetric quantum theory. Remarkably, we find larger
classes of symmetry transformations than those assumed before. This suggests a
possible direction for search of extensions of known physics.
