  A class of vector-tensor theories arises naturally in the framework of
quadratic gravity in spacetimes with linear vector distortion. Requiring the
absence of ghosts for the vector field imposes an interesting condition on the
allowed connections with vector distortion: the resulting one-parameter family
of connections generalises the usual Weyl geometry with polar torsion. The
cosmology of this class of theories is studied, focusing on isotropic solutions
wherein the vector field is dominated by the temporal component. De Sitter
attractors are found and inhomogeneous perturbations around such backgrounds
are analysed. In particular, further constraints on the models are imposed by
excluding pathologies in the scalar, vector and tensor fluctuations. Various
exact background solutions are presented, describing a constant and an evolving
dark energy, a bounce and a self-tuning de Sitter phase. However, the latter
two scenarios are not viable under a closer scrutiny.
