I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference between different valencies when it comes to higher order tensors.

I have this picture in my mind for the lower order tensors

$X^i = \left(\begin{array}{x} x^1 \\\\ x^2 \\\\ x^3\end{array}\right)$

$X_i = \left(\begin{array}{ccc} x_1 & x_2 & x_3\end{array}\right)$

$X^i_j = \left(\begin{array}{ccc} x^1_1 & x^1_2 & x^1_3 \\\\ x^2_1 & x^2_2 & x^2_3 \\\\ x^3_1 & x^3_2 & x^3_3\end{array} \right)$

for $X^{ij}$ and $X_{ij}$ they are represented in the same 2d array, but the action on a vector isn't defined in the same way as with matrices.

What I am having trouble with is intuitively understanding the difference between $X^{ijk}$, $X_{k}^{ij}$, $X_{jk}^{i}$, and $X_{ijk}$ (other permutations of the valence $(2,1)$ and $(1,2)$ omitted for brevity).

**ADDED**
After reading the responses and their comments I came up with this new picture in my head for higher order tensors.

Since I am somewhat comfortable with tensor products in quantum mechanics, I can draw a parallel with the specific tensor space I'm used to.

If we consider a rank-5 tensor with a valence of (2,3) then we can consider it in the braket notation as

$ \langle \psi_i \mid \otimes \ \langle \psi_j \mid \otimes \ \langle \psi_k \mid \otimes \mid \psi_l \rangle \ \otimes \mid \psi_m \rangle = X_{ijk}^{lm} $

Now if we operate with this tensor on rank-3 contravariant tensor, we are-left with a constant (from the inner product) and a rank-2 contravariant tensor, unmixed tensor product $\begin{eqnarray}(\langle \psi_i \mid \otimes \ \langle \psi_j \mid \otimes \ \langle \psi_k \mid \otimes \mid \psi_l \rangle \ \otimes \mid \psi_m \rangle)(\mid \Psi_i \rangle \ \otimes \mid \Psi_j \rangle \ \otimes \mid \Psi_k \rangle) &=& c \mid \psi_l \rangle \ \otimes \mid \psi_m \rangle \\\\ &=& X_{ijk}^{lm}\Psi^{ijk} = cX'^{lm}\end{eqnarray}$

If we were to further operate with a rank-2 covariant tensor (from the right, per convention that a covector and vector facing each other is an implied direct product) we would simply get a number out.

One thing I am confused about though, is that in one of the answer to this question there was a point made that we are taking tensor products of a Vector space with itself (and possibly it's dual), however in the quantum mechanics picture (although I didn't rely on it in this example) we often take tensor products between different, often disjoint, subspaces of the enormous Hilbert space that describes the quantum mechanical universe. Does the tensor picture change in this case?

Any comments on my example would be appreciated.