I am recently learning state space representation of Single-Input Single-Output LTI system, by the book Modern Control Engineering, 5th edition.
Suppose a system is represented by a homogeneous transfer function $G(s)=\frac{N(s)}{D(s)}=\frac{b_0s^n+b_1s^{n-1}+...+b_{n-1}s+b_n}{s^n+a_1s^{n-1}+...+a_{n-1}s+a_n}$, whose state space model(controllable canonical form) is illustrated in the following figure,
where state variables are given by 
The system can also be interpreted into observable canonical form, as the figure follows,
and state variables respectively,
I have my own understanding (maybe wrong) of controllable canonical form, that the scalar $u(t)$ is inputted into the highest derivative of state variable, which drives the dynamics of the system. An observer (which is the numerator $N(s)$ of the transfer function) watches every order of derivatives and put a weight onto them, thus forms the output. The observer does not change the dynamics of the system. I noticed that the input of CCF goes through a cascaded train of with a serial of integrators while the output is parallel, but in OCF the output goes out from serial integrators and the input goes into every integrator parallelly.
I am looking for the duality of CCF and OCF, by the 'findings' above, I thought if I inverse the input and output of a CCF, the system turns into OCF, with sums (nodes) turns into separators, and separators turns into sums. Does this makes any sense? How could I understand the duality?
So, let us start with the following representation of a SISO linear system with rational transfer function $H(s)$ and consider a minimal realization $(A,B,C,D)$, that is, $H(s)=C(sI-A)^{-1}B+D$. We can write the associated state-space system
$$ \begin{array}{rcl} \dot{x}(t)&=&Ax(t)+Bu(t)\\ y(t)&=&Cx(t)+Du(t). \end{array} $$
Since $H(s)$ is a scalar, then have that $H(s)=H(s)^T$ and, therefore
$$H(s)=(C(sI-A)^{-1}B+D)^T=B^T(sI-A^T)^{-1}C^T+D^T.$$
This means that the following state-space system $$ \begin{array}{rcl} \dot{\tilde x}(t)&=&A^T\tilde x(t)+C^Tu(t)\\ y(t)&=&B^T\tilde x(t)+D^T u(t). \end{array} $$ describes the same input-output relationship, but in a different state-space. Let us call that the dual system.
If we consider now the controllable canonical form $(A_c,B_c,C_c,D_c)$ and the observable canonical form $(A_o,B_o,C_o,D_o)$ as defined in your post, we can observe that
$$A_c=A_o^T,\ B_c=C_o^T,\ C_c=B_o^T,\ \textrm{and } D_c=D_o^T.$$
So, according to our definition, the controllable form is the dual of the observable form, and vice-versa.
That said, we can now look at your "flow interpretation". If you look closely, the fact of transposing and swapping the matrices have the following effect on the flow diagram: