  The aim of this work is the mathematical analysis of the physical
time-reversal operator and its definition as a geometrical structure\QTR{bf}{,
}in such a way that it could be generalized to the purely mathematical realm.
Rigorously, only having such a ``time-reversal structure'' it can be decided
whether a dynamical system is time-symmetric or not.\QTR{it}{\}The
``time-reversal structures'' of several important physical and mathematical
examples are presented, showing that there are some mathematical categories
whose objects are the (classical or abstract) ``time-reversal systems'' and
whose morphisms generalize the Wigner transformation.
