  For the evolution of a closed surface under anisotropic surface diffusion
with a general anisotropic surface energy $\gamma(\boldsymbol{n})$ in three
dimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, by
introducing a novel symmetric positive definite surface energy matrix
$\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function
$k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbol{\xi}$-vector, we present
a new symmetrized variational formulation for anisotropic surface diffusion
with weakly or strongly anisotropic surface energy, which preserves two
important structures including volume conservation and energy dissipation. Then
we propose a structural-preserving parametric finite element method (SP-PFEM)
to discretize the symmetrized variational problem, which preserves the volume
in the discretized level. Under a relatively mild and simple condition on
$\gamma(\boldsymbol{n})$, we show that SP-PFEM is unconditionally energy-stable
for almost all anisotropic surface energies $\gamma(\boldsymbol{n})$ arising in
practical applications. Extensive numerical results are reported to demonstrate
the efficiency and accuracy as well as energy dissipation of the proposed
SP-PFEM for solving anisotropic surface diffusion in 3D.
