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We construct a planar diffusion process whose infinitesimal generator dependsonly on the order of the components of the process. Speaking informally and abit imprecisely for the moment, imagine you run two Brownian-like particles onthe 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 itsrealization in terms of appropriate systems of stochastic differentialequations, study its dynamics under a time reversal, and note that theseinvolve singularly continuous components governed by local time. Crucial in ouranalysis are properties of Brownian and semimartingale local time; propertiesof the generalized perturbed Tanaka equation which we study here in detail; andthose of a one-dimensional diffusion with bang-bang drift.  We also show that our planar diffusion can be represented in terms of aprocess with bang-bang drift, its local time at the origin, and an independentstandard Brownian motion, in a form which can be construed as a two-dimensionalanalogue of the stochastic equation satisfied by the so-called skew Brownianmotion.

Planar Diffusions with Rank-Based Characteristics: Transition Probabilities, Time Reversal, Maximality and Perturbed Tanaka equations