Any quasi-isometry of the curve complex is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.
Kasra Rafi. Saul Schleimer. "Curve complexes are rigid." Duke Math. J. 158 (2) 225 - 246, 1 June 2011. https://doi.org/10.1215/00127094-1334004