Invariants and equidecomposability in rings of polygons with sides of given directions

We investigate equidecomposability in the ring of polygons with sides restricted to given directions and using only translations. Extending classical results of Dehn and Hadwiger, we prove that equidecomposability in these rings is equivalent to the equality of some translation invariant simple valuations. We also consider the algebraic structure of direction sets. We show that under mild conditions, equidecomposability with respect to a set $S$ of slopes of the given directions is equivalent to equidecomposability with respect to the field generated by $S$. We also provide a complete description of all invariants of these polygon rings.