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Jan 26, 2015 at 16:19 comment added Ryan Budney Small comment: one can arrange that the gluing maps from the 1st tetrahedron to the 2nd are the ones that send the triangle opposite vertex $i$ to the triangle opposite vertex $i+2 \ mod \ 4$. Moreover, on those triangles the gluing map is the one that preserves the order of the vertices (coming from their labelling in the ambient tetrahedron with vertices $\{0,1,2,3\}$). So if you switch the two tetrahedra around (identity on vertices), it descends to an involution of the underlying manifold. One still has to check that there's no fixed points on the interior of the edges or triangles.
Nov 18, 2011 at 22:12 history answered Igor Rivin CC BY-SA 3.0