Assuming causality and zero initial condition, the output of the cascade of an integrator and a differentiator is the same regardless of order. If we have integrator first, differentiator second, then
$$y (t) = \frac{\mathrm d}{\mathrm d t} \int_0^t u (\tau) \, \mathrm d \tau = u (t)$$
If we have differentiator first, integrator second, then
$$y (t) = \int_{0^-}^t \dot u (\tau) \, \mathrm d \tau = u (t) - \underbrace{u (0^-)}_{= 0} = u (t)$$
where the Fundamental Theorem of Calculus was used in both cases. Note that the cascade of an integrator and a differentiator produces an identity system: the output of the cascade is equal to the input of the cascade, i.e., $y (t) = u (t)$. Thus, we say that the differentiator and the integrator are the inverse systems of each other.
Assuming causality and nonzero initial condition, then things are more interesting:
If we have integrator first, differentiator second, then the initial condition of the integrator is annihilated by the differentiator and we have an identity system.
If we have differentiator first, integrator second, then the initial condition of the integrator is not annihilated by the differentiator and the output of the cascade will be the input plus a constant.
The crux of the matter is that it is the output of the cascade that matters, not the output of the first block in the cascade.