  Degenerate scalar-tensor theories are recently proposed covariant theories of
gravity coupled with a scalar field. Despite being characterised by higher
order equations of motion, they do not propagate more than three degrees of
freedom, thanks to the existence of constraints. We discuss a geometrical
approach to degenerate scalar-tensor systems, and analyse its consequences. We
show that some of these theories emerge as a certain limit of DBI Galileons. In
absence of dynamical gravity, these systems correspond to scalar theories
enjoying a symmetry which is different from Galileon invariance. The scalar
theories have however problems concerning the propagation of fluctuations
around a time dependent background. These issues can be tamed by breaking the
symmetry by hand, or by minimally coupling the scalar with dynamical gravity in
a way that leads to degenerate scalar-tensor systems. We show that distinct
theories can be connected by a relation which generalizes Galileon duality, in
certain cases also when gravity is dynamical. We discuss some implications of
our results in concrete examples. Our findings can be helpful for assessing
stability properties and understanding the non-perturbative structure of
systems based on degenerate scalar-tensor systems.
