We investigate the phase space symmetries and conserved charges of
homogeneous gravitational minisuperspaces. These (0+1)-dimensional reductions
of general relativity are defined by spacetime metrics in which the dynamical
variables depend only on a time coordinate, and are formulated as mechanical
systems with a non-trivial field space metric (or supermetric) and effective
potential. We show how to extract conserved charges for those minisuperspaces
from the homothetic Killing vectors of the field space metric. In the case of
two-dimensional field spaces, we exhibit a universal 8-dimensional symmetry
algebra given by the semi-direct sum of
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$ with the two-dimensional
Heisenberg algebra $\mathfrak{h}_2\simeq\mathbb{R}^4$. We apply this to the
systematic study of the Bianchi models for homogeneous cosmology. This extends
previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for
Friedmann-Lemaitre-Robertson-Walker cosmology, and the Poincar\’e symmetry for
Kantowski-Sachs metrics describing the black hole interior. The presence of
this rich symmetry structure already in minisuperspace models opens new doors
towards quantization and the study of solution generating mechanisms.
