Symmetry of the $3times 3$ Cauchy Stress Tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a surface), and returns a vector $t_n$ which is the traction vector. It is then shown that conservation of angular momentum leads to symmetry of the matrix.

However, tensors are more more naturally presented a multilinear functions. I wonder:

• What type of tensor is the Cauchy stress tensor? Is $n$ a vector or a co-vector? What about $t_n$?


• Is there a way to understand the symmetry when thinking of the stress tensor as a function of two vectors (or two co-vectors), under which it will seems intuitive why $\sigma(A,B) = \sigma(B,A)$?

Edit: To clarify, let's look, for example, at the 1st coordinate of the traction vector $t_n$ of an arbitrary normal $n$: This is $\left$. From symmetry, this is equivalent to $\left$ - the inner product of $n$ with the traction vector of a surface orthogonal to $e_1$. Mathematically, I understand why this is correct. But is there any intuitive meaning as to why these two quantities are the same?