  We build upon previous investigation of the one-dimensional conformal
symmetry of the Friedman-Lema\^ itre-Robertson-Walker (FLRW) cosmology of a
free scalar field and make it explicit through a reformulation of the theory at
the classical level in terms of a manifestly
$\textrm{SL}(2,\mathbb{R})$-invariant action principle. The new tool is a
canonical transformation of the cosmological phase space to write it in terms
of a spinor, i.e. a pair of complex variables that transform under the
fundamental representation of $\textrm{SU}(1,1)\sim\textrm{SL}(2,\mathbb{R})$.
The resulting FLRW Hamiltonian constraint is simply quadratic in the spinor and
FLRW cosmology is written as a Schr\"odinger-like action principle. Conformal
transformations can then be written as proper-time dependent
$\textrm{SL}(2,\mathbb{R})$ transformations. We conclude with possible
generalizations of FLRW to arbitrary quadratic Hamiltonian and discuss the
interpretation of the spinor as a gravitationally-dressed matter field or
matter-dressed geometry observable.
