  We explore the implications of hidden symmetries present in a particular
quantum cosmological setting, extending the results reported in \cite{10,11}.
In more detail, our case study is constituted by a spatially closed
Friedmann-Lema\^{\i}tre-Robertson-Walker universe, in the presence of a
conformally coupled scalar field. The $su(1,1)$ hidden symmetries of this
model, together with the Hamiltonian constraint, lead to the gauge invariance
of its corresponding Bargmann indices. We subsequently show that some
factor-ordering choices can be related to the allowed spectrum of Bargmann
indices and hence, to the hidden symmetries. Moreover, the presence of those
hidden symmetries also implies a set of appropriate boundary conditions to
choose from. In summary, our results suggest that factor ordering and boundary
conditions can be intertwined when a quantum cosmological model has hidden
symmetries.
