  Horava gravity has been proposed as a renormalizable quantum gravity without
the ghost problem through anisotropic scaling dimensions which break Lorentz
symmetry in UV. In the Hamiltonian formalism, due to the Lorentz-violating
terms, the constraint structure looks quite different from that of general
relativity (GR) but we have recently found that "there exists the case where we
can recover the same number of degrees of freedom as in GR", in a rather
general set-up. In this paper, we study its Lagrangian perspectives and examine
the full diffeomorphism (Diff) symmetry and its associated conservation laws in
Horava gravity. Surprisingly, we find that the full Diff symmetry in the action
can also be recovered when a certain condition, called "super-condition", which
super-selects the Lorentz-symmetric sector in Horava gravity, is satisfied.
This indicates that the broken Lorentz symmetry, known as
"foliation-preserving" Diff, is just an "apparent" symmetry of the Horava
gravity action and rather its "full action symmetry can be as large as the Diff
in GR ". The super-condition exactly corresponds to the tertiary constraint in
Hamiltonian formalism which is the second-class constraint and provides a
non-trivial realization of the Lorentz symmetry otherwise being absent
apparently. From the recovered Lorentz symmetry in the action, we obtain the
conservation laws with the Noether currents as in covariant theories. The
general formula for the conserved Noether charges reproduces the mass of
four-dimensional static black holes with an "arbitrary" cosmological constant
in Horava gravity, and is independent of ambiguities associated with the choice
of asymptotic boundaries. We also discuss several challenging problems,
including its implications to Hamiltonian formalism, black hole thermodynamics,
radiations from colliding black holes.
