  We derive the isometry irrep content of squashed seven-sphere
compactifications of eleven-dimensional supergravity, i.e., the left-squashed
($LS^7$) with ${\mathcal N}=1$ and right-squashed ($RS^7$) with ${\mathcal
N}=0$ supersymmetry, in a manner completely independent of the round sphere.
Then we compare this result with the spectrum obtained by Higgsing the round
sphere spectrum. This way we discover features of the spectra which makes it
possible to argue that the only way the round spectrum can be related by a
Higgs mechanism to the one of $LS^7$ is if the singletons are included in the
round sphere spectrum. For this to work also in the $RS^7$ case it seems that
the gravitino of the $LS^7$ spectrum must be replaced by a fermionic singleton
present in the $RS^7$ spectrum.
