Neutral is not your friend
First I'd address the common misconception that neutral is harmless somehow, which probably arises from people seeing neutral wires bonded to ground wires, therefore same, therefore safe. Nope.
Hot and neutral are the active conductors meaning current normally flows on them. This means neutral is a hot wire and you should fear it. One might say neutral is near ground when everything is working normally, but that fact is rather irrelevant to those of us opening up boxes or panels, since we wouldn't be in there if everything was working normally.
So yes, expect neutral to be nearly as bitey as its hot sister.
GFCIs pipe hot and neutral through themselves
Here is a schematic of how GFCIs work.
Here's how that works. The GFCI breaker clips onto the "bus" for supply hot, in the usual way. For neutral, it has a "pigtail" to the neutral bar. That pigtail is deliberately coiled up to make it distinctive and notable, to help you distinguish it from other wires in the panel. That takes care of the black and white in the diagram.
Now I deliberately colored the "protected load" wires differently to reflect that they must be kept separate. However yours are probably black and white because cables are made that way. But it's pretty easy; the hot and neutral from the Romex simply go to the hot and neutral terminals on the breaker. There's no need to even separate them, really; just run the black and white together.
Anway, as you can see, that neutral pigtail (the white wire in the chart above) powers both the GFCI and the entire circuit. Therefore one should expect series arcing when disconnecting it with the circuit live. Further, the neutral wire will have hot voltage on it, because like I say, it's a hot. Under ideal conditions, it would be near ground, but all bets are off when you remove a wire.
You mentioned "neat" and that worries me that you've had a visit from a fellow named Captain Snippy. This fellow nips every wire as short as it can possibly be. His most cherished tool is his wire cutter, and he has a shrine of all his cut off wire bits. I have a feeling the Captain has a bit of OCD, and is strictly following some internal rule in his brain.
Here is the internal rule in my brain: All hot and neutral wires must be long enough to reach every breaker space in the panel. Neutral because of GFCI and AFCI, and every breaker space so you or the next person is able to rearrange breaker positions for unforeseeable future needs such as gen interlock, surge suppressor, GFCI/AFCI, stab limits, logical organization, you name it.
Unfortunately if you're following in the wake of Captain Snippy, you soon end up with a panel full of wire nuts.