  We consider polygons with the following ``pairing property'': for each edge
of the polygon there is precisely one other edge parallel to it. We study the
problem of when such a polygon $K$ tiles the plane multiply when translated at
the locations $\Lambda$, where $\Lambda$ is a multiset in the plane. The
pairing property of $K$ makes this question particularly amenable to Fourier
Analysis. After establishing a necessary and sufficient condition for $K$ to
tile with a given lattice $\Lambda$ (which was first found by Bolle for the
case of convex polygons-notice that all convex polygons that tile, necessarily
have the pairing property and, therefore, our theorems apply to them) we move
on to prove that a large class of such polygons tiles only quasi-periodically,
which for us means that $\Lambda$ must be a finite union of translated
2-dimensional lattices in the plane. For the particular case of convex polygons
we show that all convex polygons which are not parallelograms tile necessarily
quasi-periodically, if at all.
