  Hohlweg and Lange (2007) and Santos (2004, unpublished) have found two
different ways of constructing exponential families of realizations of the
n-dimensional associahedron with normal vectors in {0,1,-1}^n, generalizing the
constructions of Loday (2004) and Chapoton-Fomin-Zelevinsky (2002). We classify
the associahedra obtained by these constructions modulo linear equivalence of
their normal fans and show, in particular, that the only realization that can
be obtained with both methods is the Chapoton-Fomin-Zelevinsky (2002)
associahedron.
  For the Hohlweg-Lange associahedra our classification is a priori coarser
than the classification up to isometry of normal fans, by
Bergeron-Hohlweg-Lange-Thomas (2009). However, both yield the same classes. As
a consequence, we get that two Hohlweg-Lange associahedra have linearly
equivalent normal fans if and only if they are isometric.
  The Santos construction, which produces an even larger family of
associahedra, appears here in print for the first time. Apart of describing it
in detail we relate it with the c-cluster complexes and the denominator fans in
cluster algebras of type A.
  A third classical construction of the associahedron, as the secondary
polytope of a convex n-gon (Gelfand-Kapranov-Zelevinsky, 1990), is shown to
never produce a normal fan linearly equivalent to any of the other two
constructions.
