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We present observations with the Chandra X-ray Observatory of the pulsar wind nebula (PWN) powered by the energetic young pulsar B1509-58. These data confirm the complicated morphology of the system indicated by previous observations, and in addition reveal several new components to the nebula. The overall PWN shows a clear symmetry axis oriented at a position angle 150 degrees (N through E), which we argue corresponds to the pulsar spin axis. We show that a previously identified radio feature matches well with the overall extent of the X-ray PWN, and propose the former as the long-sought radio nebula powered by the pulsar. We further identify a bright collimated feature, at least 4' long, lying along the nebula's main symmetry axis; we interpret this feature as a physical outflow from the pulsar, and infer a velocity for this jet >0.2c. The lack of any observed counter-jet implies that the pulsar spin axis is inclined at approx 30 deg to the line-of-sight, contrary to previous estimates made from lower-resolution data. We also identify a variety of compact features close to the pulsar. A pair of semi-circular X-ray arcs lie 17 and 30 arcsec to the north of the pulsar; the latter arc shows a highly-polarized radio counterpart. We show that these features can be interpreted as ion-compression wisps in a particle-dominated equatorial flow, and use their properties to infer a ratio of electromagnetic to particle energy in pairs at the wind shock sigma approx 0.005, similar to that seen in the Crab Nebula. We further identify several compact knots seen very close to the pulsar; we use these to infer sigma < 0.003 at a separation from the pulsar of 0.1 pc.