  This article demonstrates that additionally to the well-known velocity memory
effect, a vacuum gravitational plane wave can also induce a displacement memory
on a couple of test particles. A complete classification of the conditions
under which a velocity or a displacement memory effect occur is established.
These conditions depend both the initial conditions of the relative motion and
on the wave profile. The two cases where the wave admits a pulse or a step
profile are treated. Our analytical expressions are then compared to numerical
integrations to exhibit either a velocity or a displacement memory, in the case
of these two families of profiles. Additionally to this classification, the
existence of a new symmetry of polarized vacuum gravitational plane wave under
M\"{o}bius reparametrization of the null time is demonstrated. Finally, we
discuss the resolution of the geodesic deviation equation by means of the
underlying symmetries of vacuum gravitational plane wave.
