  Certain quantum gates, such as the controlled-NOT gate, are symmetric in
terms of the operation of the control system upon the target system and vice
versa. However, no operational criteria yet exist for establishing whether or
not a given quantum gate is symmetrical in this sense. We consider a
restricted, yet broad, class of two-party controlled gate operations for which
the gate transforms a reference state of the target into one of an orthogonal
set of states. We show that for this class of gates it is possible to establish
a simple necessary and sufficient condition for the gate operation to be
symmetric.
