  We construct a two-dimensional diffusion process with rank-dependent local
drift and dispersion coefficients, and with a full range of patterns of
behavior upon collision that range from totally frictionless interaction, to
elastic collision, to perfect reflection of one particle on the other. These
interactions are governed by the left- and right-local times at the origin for
the distance between the two particles. We realize this diffusion in terms of
appropriate, apparently novel systems of stochastic differential equations
involving local times, which we show are well posed. Questions of pathwise
uniqueness and strength are also discussed for these systems.
  The analysis depends crucially on properties of a skew Brownian motion with
two-valued drift of the bang-bang type, which we also study in some detail.
These properties allow us to compute the transition probabilities of the
original planar diffusion, and to study its behavior under time reversal.
