  We show that links of invertible polynomials coming from the Johnson and
Koll\'ar list of K\"ahler-Einstein 3-folds that are rational homology 7-spheres
remain rational homology 7-spheres under the so-called Berglund-H\"ubsch
transpose rule coming from classical mirror symmetry constructions. Actually,
this rule produces twins, that is, links with same degree, Milnor number and
homology H_3, with the exception of iterated Thom-Sebastiani sums of
singularities of chain and cycle type, where the torsion and the Milnor number
may vary. The Berglund-H\"ubsch transpose rule not only gives a framework to
better understand the existence of SasakiEinstein twins but also gives a
mechanism for producing new examples of Sasaki-Einstein twins in the rational
homology 7 -sphere setting. We also give reasonable conditions for a
Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the
BH-transpose rule. In particular, we found 75 new examples of Sasaki-Einstein
rational homology 7-spheres arising as links of not well-formed hypersurface
singularities.
