  In earlier papers, we introduced spherical T-duality, which relates pairs of
the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\rightarrow M$ and a
7-cocycle $H$ on $P$ called the 7-flux. Intuitively, the spherical T-dual is
another such pair $(\hat P, \hat H)$ and spherical T-duality exchanges the
7-flux with the Euler class, upon fixing the Pontryagin class and the second
Stiefel-Whitney class. Unless $\mathrm{dim}(M)\leq 4$, not all pairs admit
spherical T-duals and the spherical T-duals are not always unique. In this
paper, we define a canonical Poincar\'e virtual line bundle $\mathcal{P}$ on
$S^3 \times S^3$ (actually also for $S^n\times S^n$) and the spherical
Fourier-Mukai transform, which implements a degree shifting isomorphism in
K-theory on the trivial $S^3$-bundle. This is then used to prove that all
spherical T-dualities induce natural degree-shifting isomorphisms between the
7-twisted K-theories of the pairs $(P,H)$ and $(\hat P, \hat H)$ when
$\mathrm{dim}(M)\leq 4$, improving our earlier results.
