  We describe the structure of the free actions of the Klein bottle group by
orientation preserving homeomorphisms of the plane. This group is generated by
two elements $a,b$, where the conjugate of $b$ by $a$ equals the inverse of
$b$. The main result is that $a$ must act properly discontinuously, while $b$
cannot act properly discontinuously. As a corollary, we describe some torsion
free groups that cannot act freely on the plane. We also find some properties
which are reminiscent of Brouwer theory for the infinite cyclic group $Z$, in
particular that every free action is virtually wandering.
