Code & order in polygonal billiards

Two polygons $P,Q$ are code equivalent if there are billiard orbits $u,v$ which hit the same sequence of sides and such that the projections of the orbits are dense in the boundaries $\mathrm{\partial }P,\mathrm{\partial }Q$. Our main results show when code equivalent polygons have the same angles, resp. are similar, resp. affinely similar.