  We introduce spherical T-duality, which relates pairs of the form $(P,H)$
consisting of a principal $SU(2)$-bundle $P\rightarrow M$ and a 7-cocycle $H$
on $P$. Intuitively spherical T-duality exchanges $H$ with the second Chern
class $c_2(P)$. Unless $dim(M)\leq 4$, not all pairs admit spherical T-duals
and the spherical T-duals are not always unique. Nonetheless, we prove that all
spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted
cohomologies of the bundles and, when $dim(M)\leq 7$, also their integral
twisted cohomologies and, when $dim(M)\leq 4$, even their 7-twisted K-theories.
While spherical T-duality does not appear to relate equivalent string theories,
it does provide an identification between conserved charges in certain distinct
IIB supergravity and string compactifications.
