  We introduce a large class of scalar-tensor models with interactions
containing the second derivatives of the scalar field but not leading to
additional degrees of freedom. These models exhibit peculiar features, such as
an essential mixing of scalar and tensor kinetic terms, which we have named
kinetic braiding. This braiding causes the scalar stress tensor to deviate from
the perfect-fluid form. Cosmology in these models possesses a rich
phenomenology, even in the limit where the scalar is an exact Goldstone boson.
Generically, there are attractor solutions where the scalar monitors the
behaviour of external matter. Because of the kinetic braiding, the position of
the attractor depends both on the form of the Lagrangian and on the external
energy density. The late-time asymptotic of these cosmologies is a de Sitter
state. The scalar can exhibit phantom behaviour and is able to cross the
phantom divide with neither ghosts nor gradient instabilities. These features
provide a new class of models for Dark Energy. As an example, we study in
detail a simple one-parameter model. The possible observational signatures of
this model include a sizeable Early Dark Energy and a specific equation of
state evolving into the final de-Sitter state from a healthy phantom regime.
