  (Abridged) We present high-resolution Chandra X-ray observations of PSR
B0833-45, the 89 ms pulsar associated with the Vela supernova remnant. We have
acquired two observations separated by one month to search for changes in the
pulsar and its environment following an extreme glitch in its rotation
frequency. We find a well-resolved nebula with a toroidal morphology remarkably
similar to that observed in the Crab Nebula, along with an axial Crab-like jet.
Between the two observations the flux from the pulsar is found to be steady to
within 0.75%; the 3 sigma limit on the fractional increase in the pulsar's
X-ray flux is < ~10^-5 of the inferred glitch energy. We use this limit to
constrain parameters of glitch models and neutron star structure. We do find a
significant increase in the flux of the nebula's outer arc; if associated with
the glitch, the inferred propagation velocity is > 0.7c, similar to that seen
in the brightening of the Crab Nebula wisps. We propose an explanation for the
X-ray structure of the Vela synchrotron nebula based on a model originally
developed for the Crab Nebula. In a departure from the Crab model, the
magnetization parameter "sigma" of the Vela pulsar wind is allowed to be of
order unity; this is consistent with the simplest MHD transport of magnetic
field from the pulsar to the nebula, where B < 4 X 10^-4 G. We review effects
that may enhance the probability of alignment between the spin axis and space
velocity of a pulsar, and speculate that short-period, slowly moving pulsars
are just the ones best-suited to producing synchrotron nebulae with such
aligned structures.
