  We describe recent Chandra ACIS observations of the Vela-like pulsar PSR
J2021+3651 and its pulsar wind nebula (PWN). This `Dragonfly Nebula' displays
an axisymmetric morphology, with bright inner jets, a double-ridged inner
nebula, and a ~30" polar jet. The PWN is embedded in faint diffuse emission: a
bow shock-like structure with standoff ~1' brackets the pulsar to the east and
emission trails off westward for 3-4'. Thermal (kT=0.16 +/-0.02 keV) and power
law emission are detected from the pulsar. The nebular X-rays show spectral
steepening from Gamma=1.5 in the equatorial torus to Gamma=1.9 in the outer
nebula, suggesting synchrotron burn-off. A fit to the `Dragonfly' structure
suggests a large (86 +/-1 degree) inclination with a double equatorial torus.
Vela is currently the only other PWN showing such double structure. The >12 kpc
distance implied by the pulsar dispersion measure is not supported by the X-ray
data; spectral, scale and efficiency arguments suggest a more modest 3-4 kpc.
