The authors classify all homeomorphisms of the double cover of the Sierpiński gasket in \(n\) dimensions. They show that there is a unique homeomorphism mapping any cell to any other cell with a prescribed mapping of boundary points, and any homeomorphism is either a permutation of a finite number of topological cells or a mapping of infinite order with one or two fixed points. In contrast, they show that any compact fractafold based on the level-3 Sierpiński gasket is topologically rigid.