  The Kerr geometry admits the Carter symmetry, which ensures that the geodesic
equations are integrable. It is shown that gravitational waveforms associated
with extreme-mass-ratio inspirals involving a non-integrable compact object
display `glitch' phenomena, where the frequencies of gravitational waves
increase abruptly, when the orbit crosses certain spacetime regions known as
Birkhoff islands. The presence or absence of these features in data from
upcoming space-borne detectors will therefore allow not only for tests of
general relativity, but also of fundamental spacetime symmetries.
