  Factorization structures occur in toric differential and discrete geometry,
and can be viewed in multiple ways, e.g., as objects determining substantial
classes of explicit toric Sasaki and K\"ahler geometries, as special
coordinates on such, or as an apex generalisation of cyclic polytopes featuring
a generalised Gale's evenness condition. This article presents a comprehensive
study of factorization structures. It establishes their structure theory and
introduces their use in the geometry of cones and polytopes. The article
explains the construction of polytopes and cones compatible with a given
factorization structure, and exemplifies it for product Segre-Veronese and
Veronese factorization structures, where the latter case includes cyclic
polytopes. Further, it derives the generalised Gale's evenness condition for
compatible cones, polytopes and their duals, and explicitly describes faces of
these. Factorization structures naturally provide generalised Vandermonde
identities, which relate normals of any compatible polytope, and which are used
for Veronese factorization structure to find examples of Delzant and rational
Delzant compatible polytopes. The article offers a myriad of factorization
structure examples, which are later characterised to be precisely factorization
structures with decomposable curves, and raises the question if these encompass
all factorization structures, i.e., the existence of an indecomposable
factorization curve.
