  The relativistic aberration of a wavevector and the corresponding Doppler
shift are examined in connection with superluminal and subluminal
spatiotemporally localized pulsed optical waves. The requirement of a null
Doppler shift is shown to give rise to a speed associated with the relativistic
velocity composition law of a double (two-step) Lorentz transformation. The
effects of such a transformation are examined both in terms of four-coordinates
and in the spectral domain. It is established that a subluminal pulse reverses
its direction. In addition to a change in direction, the propagation term of a
superluminal pulse becomes negative. The aberration due to a double Lorentz
transformation is examined in detail for propagation invariant superluminal
waves (X wave, Bessel X wave), as well as intensity-invariant superluminal and
subluminal waves. Detailed symmetry considerations are provided for the
superluminal focus X wave and the subluminal MacKinnon wavepacket.
