Trapped in a hyper cube. Our role playing group got stuck in a hypercube.
I recognized it as a hypercube, when after travelling "south" 4 times, I ended up in a loop, and counted 8 rooms.
After the game the GM showed me this map 
If I swapped the NS EW UP pairs, for NE WU PS pairs, would the resulting shape/network still represent a hypercube? if not, what would the new shape be called? Or would it no longer make any sense geometrically?
Each room we are in had an exit to the NSEWUP . 
 A: The short answer is yes, you would still be in a hypercube.
How can you verify this? Well, the only way you can ever verify anything about the fourth dimension... by using analogies with other dimensions. Imagine now that you were a $2D$ creature walking around on a cube. 

As you can see, you would have only four directions in this case, and there would be only $6$ rooms.
The problem is, you can't really make a graph with "directions" labelling each edge, because your orientation is constantly changing. For example, if our quadrilateral friend travelled North and then East from there, he would end up in the same place as if he had just travelled East... but he would be facing in different directions when he arrived.


So when you're a being messing around in a solid of a higher dimension, it doesn't make sense to think about "directions" anymore. For this reason, your graph isn't accurate, because it is static - in reality, the "direction" in which you would travel from one room to the other would constantly change, and so the labelling of the edges would constantly change. Thus, if you made the swap of labelling that you are asking about, you would still be in the same solid - in fact, you could probably achieve that very orientation just by walking around for a little bit inside of it.
