  We revisit gravitational wave (GW) memory as the key to measuring spacetime
symmetries, extending beyond its traditional role in GW searches. In
particular, we show how these symmetries may be probed via displacement and
spin memory observations, respectively. We further find that the Einstein
Telescope's (ET) sensitivity enables constraining the strain amplitude of a
displacement memory to 2% and that of spin memory to 22%. Finally, we point out
that neglecting memory could lead to an overestimation of measurement
uncertainties for parameters of binary black hole (BBH) mergers by about 10% in
ET.
