  In this paper, we introduce a symmetrization technique for the gradient of a
$\BV$ function, which separates its absolutely continuous part from its
singular part (sum of the jump and the Cantorian part). In particular, we prove
an $\text{\emph{L}}^{\text{1}}$ comparison between the function and its
symmetrized.
  Furthermore, we apply this result to obtain Saint-Venant type inequalities
for some geometric functionals.
