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Symmetries play important roles in modern theories of physical laws. In thispaper, we review several experimental tests of important symmetries associatedwith the gravitational interaction, including the universality of free fall forself-gravitating bodies, time-shift symmetry in the gravitational constant,local position invariance and local Lorentz invariance of gravity, andspacetime translational symmetries. Recent experimental explorations forpost-Newtonian gravity are discussed, of which, those from pulsar astronomy arehighlighted. All of these tests, of very different aspects of gravity theories,at very different length scales, favor to very high precision the predictionsof the strong equivalence principle (SEP) and, in particular, generalrelativity which embodies SEP completely. As the founding principles ofgravity, these symmetries are motivated to be promoted to even stricter testsin future.

Tests of gravitational symmetries with radio pulsars