  It has been noticed that for a large class of cosmological models, the gauge
fixing of the time-reparametrization invariance does not completely fix the
clock. Instead, the system enjoys a surprising residual Noether symmetry under
a M\"{o}bius reparametrization of the proper time, which maps
gauge-inequivalent solutions to the Friedman equations onto each other. In this
work, we provide a unified treatment of this hidden conformal symmetry and its
realization in the homogeneous and isotropic sector of the
Einstein-Scalar-$\Lambda$ system. We consider the flat
Friedmann-Robertson-Walker (FRW) model, the (A)dS cosmology and provide a first
treatment of the model with spatial constant curvature. We derive the general
condition relating the choice of proper time and the conformal weight of the
scale factor, and give a detailed analysis of the conserved Noether charges
generating this physical symmetry. Our approach allows us to identify new
realizations of this symmetry while recovering previous results in a unified
manner. We also present the general mapping onto the conformal particle and
discuss the solution-generating nature of the transformations beyond the
M\"{o}bius symmetry. Finally, we show that, at least in a restricted context,
this hidden conformal symmetry is intimately related to the Kodama charges of
spherically symmetric gravity. This new connection suggests that the M\"{o}bius
invariance of cosmology is only the corner of a larger symmetry structure which
could be relevant beyond cosmological models.
