  We introduce reflexive polytopes of index l as a natural generalisation of
the notion of a reflexive polytope of index 1. These l-reflexive polytopes also
appear as dual pairs. In dimension two we show that they arise from reflexive
polygons via a change of the underlying lattice. This allows us to efficiently
classify all isomorphism classes of l-reflexive polygons up to index 200. As
another application, we show that any reflexive polygon of arbitrary index
satisfies the famous "number 12" property. This is a new, infinite class of
lattice polygons possessing this property, and extends the previously known
sixteen instances. The number 12 property also holds more generally for
l-reflexive non-convex or self-intersecting polygonal loops. We conclude by
discussing higher-dimensional examples and open questions.
