  For a half-translation surface (S,q), the associated saddle connection
complex A(S,q) is the simplicial complex where vertices are the saddle
connections on (S,q), with simplices spanned by sets of pairwise disjoint
saddle connections. This complex can be naturally regarded as an induced
subcomplex of the arc complex. We prove that any simplicial isomorphism $\phi :
A(S,q) \to A(S',q')$ between saddle connection complexes is induced by an
affine diffeomorphism $F : (S,q) \to (S',q')$. In particular, this shows that
the saddle connection complex is a complete invariant of affine equivalence
classes of half-translation surfaces. Throughout our proof, we develop several
combinatorial criteria of independent interest for detecting various geometric
objects on a half-translation surface.
