  A deep (100 ks) XMM-Newton observation of Geminga has shown two faint tails
of diffuse X-ray emission, extending for ~2' behind the pulsar, well aligned
with the proper motion (PM) direction. We report here on a recent ~20 ks
Chandra observation, which unveils a new structure, ~25'' long and ~5'' thick,
starting at the pulsar position and perfectly aligned with the PM direction,
with a surface brightness ~40 times higher than that of the XMM Tails. The
Chandra comet-like feature has a remarkably hard spectrum (photon index
0.9-1.4) and a luminosity of ~5.5x10^28 erg s^-1, comparable to the energetics
of the larger XMM one. Geminga is thus the first neutron star to show a clear
X-ray evidence of a large-scale, outer bow-shock as well as a short, inner
cometary trail.
