TITLE: Symmetries and faces of the associahedron
QUESTION [10 upvotes]: The dihedral group of order $2n+2$ acts on $K_n$, the ($n-2$)-dimensional associahedron.  Are there any other symmetries?  References?

Does the answer to 1 change if we restrict to just the 1-skeleton of $K_n$?  References?

It is "obvious" that any simple circuit (simple closed walk, simple closed path, whatever terminology you prefer) of length 4 or 5 is a 2-dimensional face of $K_n$.  Is this true?  Proof?  Reference?

REPLY [3 votes]: I have just seen this question, while looking for something else. The answer to 1 is indeed "no", and an explicit proof appears in Lemma 2.2 of Ceballos, Santos, and Ziegler - Many non-equivalent realizations of the associahedron (not surprisingly, it follows the same ideas as Patricia's).