  Operator-ordering in quantum cosmology is a major as-yet unsettled ambiguity
with not only formal but also physical consequences. We determine the
Lagrangian origin of the conformal invariance that underlies the conformal
operator-ordering choice in quantum cosmology. It is particularly naturally and
simply manifest in relationalist product-type actions (such as the Jacobi
action for mechanics or Baierlein-Sharp-Wheeler type actions for general
relativity), for which all that is required for the kinetic and potential
factors to rescale in compensation to each other. These actions themselves
mathematically sharply implementing philosophical principles relevant to
whole-universe modelling, the motivation for conformal operator-ordering in
quantum cosmology is substantially strengthened. Relationalist product-type
actions also give emergent times which amount to recovering Newtonian, proper
and cosmic time in the various relevant contexts. The conformal scaling of
these actions directly tells us how emergent time scales; if one follows suit
with the Newtonian time or the lapse in the more commonly used difference-type
Euler--Lagrange or Arnowitt--Deser--Misner type actions, one sees how these too
obey a more complicated conformal invariance. Moreover, our discovery of the
conformal scaling of the time involved permits relating how it simplifies
equations of motion with how affine parametrization simplifies geodesics.
