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

Bianchi models with a free massless scalar field: invariant sets and higher symmetries