Are there specific examples of integral polyhedra that are neither Totally Unimodular nor Total Dual Integral?
I'm only going to talk about the constraint matrix part of the problem, I don't know much about the linear system (TDI) part.

Call a matrix $A$ good if $\{x|Ax\leq b,x\geq 0\}$ is a integral polytope for all integral $b$.
TUM is precisely the set of good integral matrices. This is a result by Hoffman and Kruskal. see this for how the theorem comes about and the original paper.

If we relax the requirement that $A$ is integral, then we get a larger class of matrices. Call a good matrix $A$ trivial, if it is a scaling of a TUM matrix (i.e. $A=\frac{1}{k}B$ for some TUM matrix $B$ and integer $k$).

Sadly, I don't know of any natural non-trivial good matrices.

But there is a characterization of good matrices. A matrix $A$ is $1$-regular if for all non-singular square submatrix $R$, $R^{-1}$ is integral. We have the following by Appa and Kotnyek. 


  A rational matrix $A$ is good iff $A$ is $1$-regular.


Here is an example of a non-trivial good matrix.

$$\left(
\begin{array}{cc}
 \frac{1}{2} &amp; -\frac{1}{2} \\
 -\frac{1}{2} &amp; -\frac{1}{2} \\
\end{array}
\right)$$