  Plane gravitational waves can admit a sixth `screw' isometry beyond the usual
five. The same is true of plane electromagnetic waves. From the point of view
of integrable systems, a sixth isometry would appear to over-constrain particle
dynamics in such waves; we show here, though, that no effect of the sixth
isometry is independent of those from the usual five. Many properties of
particle dynamics in a screw-symmetric gravitational wave are also seen in a
(non-plane-wave) electromagnetic vortex; we make this connection explicit,
showing that the screw-symmetric gravitational wave is the classical double
copy of the vortex.
