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(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.