Time reversal exchanges the roles of the initial and final stages of an experiment. This fact is properly represented here by an alternative time-reversal transformation in quantum theory. In elementary quantum experiments one prepares a system, lets it propagate over time, and checks for a particular value of a complete sequence of system variables. Following the operational interpretation of quantum theory, the initial and final stages of such experiments are represented by kets and bras. Hence, the new time-reversal transformation maps kets into bras and vice versa. Wigner's result about changes of description of a quantum system is extended so as to include transformations between kets and bras. Invariance of the Schwinger action principle under time reversal requires the new time-reversal transformation to be linear. In this paper the time reversal of experiments is represented completely, whereas Wigner's formulation only applies to the propagation phase (so-called time evolution) of an experiment.