He's really wondering whether the voxel-space is a directed graph or whether up∘down=down∘up=identity (and for left/right too). Movement could be commutative with up∘down≠identity.

Consider

voxels = {a, b}
 
left(a)  = a
right(a) = a
up(a)    = a
down(a)  = a
 
left(b)  = a
right(b) = a
up(b)    = a
down(b)  = a
 
If f is in (left, right, up, down)
let g be the respective function in (right, left, down, up)
forall x in {a, b}
f(g(x))=g(f(x))=a
 
But
up(down(b))=a
whereas
identity(b)=b