Spherical symmetry arguments are used to produce a general device to convert identities and inequalities for the $p$th absolute moments of real-valued random variables into the corresponding identities and inequalities for the $p$th moments of the norms of random vectors in Hilbert spaces. Particular results include the following: (i) an expression of the $p$th moment of the norm of such a random vector $X$ in terms of the characteristic functional of $X$; (ii) an extension of a previously obtained von~Bahr--Esseen-type inequality for real-valued random variables with the best possible constant factor to random vectors in Hilbert spaces, still with the best possible constant factor; (iii) an extension of a previously obtained inequality between measures of "contrast between populations" and "spread within populations" to random vectors in Hilbert spaces.