  We reformulate the dynamics of homogeneous cosmologies with a scalar field
matter source with an arbitrary self-interaction potential in the language of
jet bundles and extensions of vector fields. In this framework, the
Bianchi-scalar field equations become subsets of the second Bianchi jet bundle,
$J^2$, and every Bianchi cosmology is naturally extended to live on a variety
of $J^2$. We are interested in the existence and behaviour of extensions of
arbitrary Bianchi-Lie and variational vector fields acting on the Bianchi
variety and accordingly we classify all such vector fields corresponding to
both Bianchi classes $A$ and $B$. We give examples of functions defined on
Bianchi jet bundles which are constant along some Bianchi models (first
integrals) and use these to find particular solutions in the Bianchi total
space. We discuss how our approach could be used to shed new light to questions
like isotropization and the nature of singularities of homogeneous cosmologies
by examining the behaviour of the variational vector fields and also give rise
to interesting questions about the `evolution' and nature of the cosmological
symmetries themselves.
