  We study gravitational waves to first and second order in amplitude in vacuum
asymptotically flat spacetimes. The Einstein equations are solved to first
order and these solutions are superposed to form a time-symmetric ingoing and
then outgoing pulse regular everywhere. The waves are assumed to have
odd-parity and a non-vanishing angular momentum which keeps them away from the
axis at all times. The averaged energy of the waves is evaluated. The relevant
Einstein equation is then solved to second order in the amplitude. The
influence of the angular momentum of the waves on the rotation of local
inertial frames with respect to the frames at great distances is analyzed. The
rotation of the frames occurs even in the region around the origin where
spacetime is almost flat. The rotation is without time delay as it follows from
the constraint equation. The results are illustrated graphically for various
values of the "harmonic index m" corresponding to azimuthal rotation and the
"harmonic index l" describing the latitudinal rotation of the waves. The
apparent motions of the fixed stars on the celestial sphere as seen through
rotating waves from the local inertial frame are calculated and displayed.
