  We consider a class of Lorentz-violating theories of gravity involving a
timelike unit vector field (the aether) coupled to a metric, two examples being
Einstein-aether theory and Ho\v{r}ava gravity. The action always includes the
Ricci scalar of the metric and the invariants quadratic in covariant
derivatives of the aether, but the theories differ in how the aether is
constructed from other fields, and whether those fields are varied in the
action. Fields that are not varied define background structures breaking
diffeomorphsim invariance, including threadings, folations, and clocks, which
generally produce novel degrees of freedom arising from the violation of what
would otherwise be initial value constraints. The principal aims of this paper
are to survey the nature of the theories that arise, and to understand the
consequences of breaking diffeomorphism invariance in this setting. In a
companion paper [arXiv:1504.03305], we address some of the phenomenology of the
"ponderable aether" case in which the presence of a background clock endows the
aether with a variable internal energy density that behaves in some respects
like dark matter.
