  In this letter, we stress that the simplest cosmological model consisting in
a massless scalar field minimally coupled to homogeneous and isotropic gravity
has an in-built $\SL(2,\mathbb{R})$ symmetry. Protecting this symmetry
naturally provides an efficient way to constrain the quantization of this
cosmological system whatever the quantization scheme and allows in particular
to fix the quantization ambiguities arising in the canonical quantization
program. Applying this method to the loop quantization of the FLRW cosmology
leads to a new loop quantum cosmology model which preserves the
$\SL(2,\mathbb{R})$ symmetry of the classical system. This new polymer
regularization consistent with the conformal symmetry can be derived as a
non-linear canonical transformation of the classical FLRW phase space, which
maps the classical singular dynamics into a regular effective bouncing
dynamics. This improved regularization preserves the scaling properties of the
volume and Hamiltonian constraint. 3d scale transformations, generated by the
dilatation operator, are realized as unitary transformations despite the
minimal length scale hardcoded in the theory. Finally, we point out that the
resulting cosmological dynamics exhibits an interesting duality between short
and long distances, reminiscent of the T-duality in string theory, with the
near-singularity regime dual to the semi-classical regime at large volume. The
technical details of the construction of this model are presented in a longer
companion paper \cite{BenAchour:2019ywl}.
