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We generalize a result of Paulin on the Gromov boundary of hyperbolic groupsto the Morse boundary of proper, maximal hierarchically hyperbolic spacesadmitting cocompact group actions by isometries. Namely we show that if theMorse boundaries of two such spaces each contain at least three points, thenthe spaces are quasi-isometric if and only if there exists a 2-stable,quasi-m\"obius homeomorphism between their Morse boundaries. Our result extendsa recent result of Charney-Murray, who prove such a classification for CAT(0)groups, and is new for mapping class groups and the fundamental groups of$3$-manifolds without Nil or Sol components.

Hierarchically hyperbolic groups are determined by their Morse boundaries