  Reparametrization invariance being treated as a gauge symmetry shows some
specific peculiarities. We study these peculiarities both from a general point
of view and on concrete examples. We consider the canonical treatment of
reparametrization invariant systems in which one fixes the gauge on the
classical level by means of time-dependent gauge conditions. In such an
approach one can interpret different gauges as different reference frames. We
discuss the relations between different gauges and the problem of gauge
invariance in this case. Finally, we establish a general structure of
reparametrizations and its connection with the zero-Hamiltonian phenomenon.
