  The results of long XMM-Newton X-ray observations of the NW radio relic of
Abell 3667 are presented. A shock is detected at the sharp outer edge of the
radio relic, both in the X-ray surface brightness and the temperature profiles.
The Mach number is M = 2.54^+0.80_-0.43. The temperature jump at the shock is
larger than expected from the density jump, which may indicate that a
dynamically important magnetic field aligned primarily parallel to the shock
front is present. The gas temperature rises gradually over several arc minutes
within the shock region. This could indicate that the shock energy is initially
dissipated into some mix of thermal and nonthermal (e.g., turbulence)
components, and that the nonthermal energy decays into heat in the post-shock
region. The observed radio relic can be powered if ~0.2% of the energy
dissipated in the shock goes into the (re)acceleration of relativistic
electrons. We show that the observed steepening of the radio spectrum with
distance behind the shock is consistent with radiative losses by the
radio-emitting electrons. However, the radio spectrum immediately behind the
shock is flatter than expected for linear diffusive shock acceleration of
thermal electrons. This suggests that the shock re-accelerates a pre-existing
population of relativistic electrons. We also detect a bright, cool region (the
"Mushroom") to the south of the radio relic, which we propose is the remnant
cool core of a merging subcluster, and that this subcluster was the driver for
the observed NW shock. In this model, the properties of Abell 3667 are mainly
the result of an offset binary merger, and the cluster is being observed about
1 Gyr after first core passage. We predict that deeper X-ray or SZ observations
of the SE radio relic will reveal a second merger shock at the outer edge.
