  The interaction between a pulsar wind nebula (PWN) and its host supernova
remnant (SNR) can produce a vast array of observable structures. Asymmetry
present within these structures derives from the complexity of the composite
system, where many factors take turns playing a dominating hand throughout the
stages of composite SNR evolution. Of particular interest are systems
characterized by blastwave expansion within a nonuniform interstellar medium
(ISM), which contain an active pulsar having a substantial "kick" velocity
(upward of 300 km s$^{-1}$ ), because these systems tend to produce complex
morphologies. We present a numerical model that employs these and several other
factors in an effort to generate asymmetry similar to that seen in various
X-ray and radio observations. We find that the main parameters driving
structure are ISM uniformity and total pulsar spin- down energy, with secondary
contributions from factors such as pulsar trajectory and initial spin-down
luminosity. We also investigate the dynamics behind PWN "tails," which may form
to link active pulsars to a crushed, relic nebula as the reverse shock passes.
We find that the directions of such tails are not good indicators of pulsar
motion, but direction does reveal the flow of ejecta created by the passage of
a reverse shock.
