  Symmetries play important roles in modern theories of physical laws. In this
paper, we review several experimental tests of important symmetries associated
with the gravitational interaction, including the universality of free fall for
self-gravitating bodies, time-shift symmetry in the gravitational constant,
local position invariance and local Lorentz invariance of gravity, and
spacetime translational symmetries. Recent experimental explorations for
post-Newtonian gravity are discussed, of which, those from pulsar astronomy are
highlighted. All of these tests, of very different aspects of gravity theories,
at very different length scales, favor to very high precision the predictions
of the strong equivalence principle (SEP) and, in particular, general
relativity which embodies SEP completely. As the founding principles of
gravity, these symmetries are motivated to be promoted to even stricter tests
in future.
