In Kingman, J. (1972) On Random Sequences with Spherical Symmetry. Biometrika, 59(2), 492-494., the author states a theorem that, in his words, "has a close resemblance to de Finetti's theorem on exchangeability".
To my best knowledge a sequence of random variables has the property of spherical symmetry if its distribution function is invariant to rotations of its elements, i.e. the distribution function evaluated in $X$ and $ΓX$ is the same given that $Γ$ is any orthogonal $nxn$ matrix.
The only thing I notice is that exchangeability is a property concerning distribution functions invariant with respect to something, i.e. permutations.
Could you please explain how these two concepts are related or different?