  Truncated Toeplitz operators are C--symmetric with respect to the canonical
conjugation given on an appropriate model space. However, by considering only
one conjugation one cannot characterize truncated Toeplitz operators. It will
be proved, for some classes of inner functions and the model spaces connected
with them, that if an operator on a model space is C--symmetric for a certain
family of conjugations in the model space, then is has to be truncated
Toeplitz. A characterization of classical Toeplitz operators is also presented
in terms of conjugations.
