  The maximally supersymmetric Freund-Rubin vacua for eleven dimensional
supergravity, namely $AdS_4 \times S^7$ and $AdS_7 \times S^4$, admit an
analytic continuation to $S^4 \times S^7$. From the full harmonic expansions on
$S^4 \times S^7$, it is shown that by analytical continuation to either
$AdS_4$, or to $AdS_7$, the detailed structure of the Kaluza-Klein spectrum can
be obtained for both vacua in a unified manner. The results are shown to be
related by a simple rule which interchanges the spacetime and internal space
representations. We also obtain the linearized field equations for the
singletons and doubletons but they can be gauged away by fixing certain
Stuckelberg shift symmetries inherited from the Kaluza-Klein reduction.
