One way to look at this....
Integrating acceleration to velocity is like passing the signal through a low pass filter, with 6 dB/octave roll-off. The higher the frequency, the less of the signal gets through.
To go the other direction accurately, that is differentiating velocity to acceleration, requires "putting back" that part of the signal that was filtered out. But there's no way to put back the high frequency information that isn't there.
The differentiated signal will lack high frequency components. Signals from impacting and from high frequency sources such as rolling element bearing deterioration and gear mesh will not be present, so the time domain signal will be distorted from the true signal.
In short, integration from acceleration to velocity to displacement is ok. Differentiation from displacement to velocity or velocity to acceleration is not so good.
Peaks from a spectrum can be accurately converted between acceleration/velocity/displacement in any direction. VibCon, a free vibration conversion calculator by yours truly, is available on CTC's web site at www.ctconline.com