  It is known that, adding the number of lattice points lying on the boundary
of a reflexive polygon and the number of lattice points lying on the boundary
of its polar, always yields 12. Generalising appropriately the notion of
reflexivity, one shows that this remains true for "$\ell$-reflexive polygons".
In particular, there exist (for this reason) infinitely many (lattice
inequivalent) lattice polygons with the same property. The first proof of this
fact is due to Kasprzyk and Nill. The present paper contains a second proof
(which uses tools only from toric geometry) as well as the description of
complementary properties of these polygons and of the invariants of the
corresponding toric log del Pezzo surfaces.
