My fist thought would be that this is related to EMI that is strong enough to bleed into the control signal wires. When you command a start, the drive pumps out its PWM into unshielded motor lead wiring, which pumps out a surge Of EMI, which gets into your digital input wires. The drive digital input circuitry has to filter it, but it gets overwhelmed for an instant, turns the drive command off or bleeds into the mP, then the filtering catches up and it goes back to normal.
The reversing is the odd man out in that, but is the drive actually reversing the motor, or is the MACHINE changing direction? Because if, let's say, this was belt driven that might explaing the appearance of reversal. For the first instant that you apply torque to a belt driven power transmission system, the belts stretches before the torque passes onto the load. If in that instant, you suddently REMOVE the torque, the belts will spring back, possibly reversing the load for a second, then when the drive reestablishes torque, the load goes forward again. Similar effect can apply to compressors, vacuum pumps and gravity dependent loads as well, (because we all know, gravity sucks) but you said the same phenomenon happens whether going Fwd or Rev, so those would be unlikely. But the fact that this happens going in either directing can be explained by belt stretch and spring back.
Here is what I would do. Take a drive and motor out of the system altogether, a set that you know for sure this happens on and put them on the bench, uncoupled to a load. Wire everything up just as you have it in the system and see if the problem repeats itself. If it does, get a piece of shielded VFD cable to go between the drive and motor and see if it cures it. If that works, there is your problem. If not, it's an issue deeper into the drive and AB needs to get involved.
If the problem is gone just from having removed the components, then you KNOW that it has something to do with the installation. So start putting things back in one at a time until the problem reveals itself, there is your effect.