  We construct a planar diffusion process whose infinitesimal generator depends
only on the order of the components of the process. Speaking informally and a
bit imprecisely for the moment, imagine you run two Brownian-like particles on
the real line. At any given time, you assign positive drift g and diffusion
{\sigma} to the laggard; and you assign negative drift -h and diffusion {\rho}
to the leader.
  We compute the transition probabilities of this process, discuss its
realization in terms of appropriate systems of stochastic differential
equations, study its dynamics under a time reversal, and note that these
involve singularly continuous components governed by local time. Crucial in our
analysis are properties of Brownian and semimartingale local time; properties
of the generalized perturbed Tanaka equation which we study here in detail; and
those of a one-dimensional diffusion with bang-bang drift.
  We also show that our planar diffusion can be represented in terms of a
process with bang-bang drift, its local time at the origin, and an independent
standard Brownian motion, in a form which can be construed as a two-dimensional
analogue of the stochastic equation satisfied by the so-called skew Brownian
motion.
