Is it possible to replace an integrator system with an equivalent differentiator?
Typically you can devise equivalent block diagrams in terms of just integrators or just differentiators.

A simple example is on page 2-15 of

https://web.stanford.edu/~boyd/ee102/systems.pdf

which should look remarkably familiar. You should be cautioned that mathematical equivalence doesn't mean that there aren't reasons where one form would be prefered over the other.  OP Amp integrators have better noise characteristics. You also have to band limit an analog differentiator to make it physically realizable. 

Another issue is that they will not in general have the same complexity.

When modeling ordinary differential equations systems, 
$$
\mathbf{M}(t) \frac{d \mathbf{y}(t)}{dt}= \mathbf{g}(t,\mathbf{y})
$$
where $\mathbf{M}(t)$ (mass matrix) is invertible is often more sparse for physical systems than 
$$
\frac{d \mathbf{y}(t)}{dt}= \mathbf{h}(t,\mathbf{y})
$$
which is a more general way of writing the common state space representation.

One can write :
$$
\frac{d \mathbf{y}(t)}{dt}= \mathbf{M}^{-1}(t)\mathbf{g}(t,\mathbf{y})
$$
but if $\mathbf{M}(t)$ is sparse, $\mathbf{M}^{-1}(t)$ probably isn't ,which makes a big difference when the dimension of $\mathbf{y}$ is large.

The Matlab Control systems toolbox also includes state variable equations of the form

$$
\mathbf{E} \frac{d \mathbf{y}}{dt}= \mathbf{A}\mathbf{y}+ \mathbf{B}\mathbf{u}$$