  We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if
and only if it is metrically quasiconformal and if and only if it is
geometrically quasiconformal. As the main step in our argument, we show that a
quasisymmetric parametrization of the Grushin plane by the Euclidean plane must
also be geometrically quasiconformal. We also discuss some aspects of the
Euclidean theory of quasiconformal maps, such as absolute continuity on almost
every compact curve, not satisfied in the Grushin case.
