  We scrutinize the overall structure of the space of cosmological models of
Bianchi type I-VII$_h$ that contain a free massless scalar field, with a
spatially homogeneous gradient $\nabla_\mu \varphi$ that generally breaks
isotropy, in addition to a standard perfect fluid. Specifically, state space is
written as a union of disjoint invariant sets, each corresponding to a
particular cosmological model that is classified with respect to the Bianchi
type and the matter content. Subsets corresponding to higher-symmetry models,
including all locally rotationally symmetric models and FLRW models, and models
with a shear-free normal congruence, are also derived and classified. For each
model the dimension ($d$) of the space of initial data is given, after fixing
the orientation of the orthonormal frame uniquely relative to matter
anisotropies and geometrical anisotropies.
