In many textbooks about the information geometry (for instance, "Information Geometry with Applications" by Amari), authors emphasize that statistical manifold is described by tripple $\{\mathcal{M},G,T\}$, where $\mathcal{M}$ is manifold, $G$ is metric on it and $T$ is third-order tensor, which is called skewness tensor. The authors also emphasize the difference between this structure and the Riemannian structure $\{\mathcal{M},G\}$.
My question: the tensor $T_{ijk}$ is symmetric, whereas in physics I know only one third-order tensor, torsion tensor, which is anti-symmetric. How should I interpret the third order symmetric tensor? What is the most suitable physical analogy for this tensor?
Additional info: statistical manifolds appear during the study of distance between probability measures. For family of probability measures $p=p(x,\xi)$, where $x$ corresponds to random variable and $\xi$ corresponds to parameters ($\xi$ is vector!), there is a unique metric tensor, defined by $$g_{ij}=\int dx\,p(x,\xi)\frac{\partial\ln p(x,\xi)}{\partial \xi_i}\frac{\partial\ln p(x,\xi)}{\partial \xi_j},$$ and there is a unique skewness tensor defined by $$T_{ijk}=\int dx\,p(x,\xi)\frac{\partial\ln p(x,\xi)}{\partial \xi_i}\frac{\partial\ln p(x,\xi)}{\partial \xi_j}\frac{\partial\ln p(x,\xi)}{\partial \xi_k}.$$
In his book, Amari introduces the third-order tensor via the concept of dual connections. Let two vectors, $A$ and $B$. Let $\langle A,B\rangle$ is the scalar product of this two vectors. Let two symmetric affince connections $\Gamma$ and $\Gamma^*$. Let $\Pi$ and $\Pi^*$ are parallel transport operations w.r.t such connections. So, we have $$\langle A,B\rangle =\langle \Pi A, \Pi^*B\rangle.$$ Next, the skewness tensor is $$T_{ijk}=\Gamma_{ijk}^*-\Gamma_{ijk}.$$
Note: I understand that this question may very specific and there are not lot of details, but it corresponds to my current understanding.