​
In[]:=
LayeredGraphPlot[GeneralizedGridGraph[{10"Directed",10"Directed"}],AspectRatio1]
Out[]=

What is this invariant under?

Coordinates of points are {t,x} or {t,x1,x2}
Isometry group:
What matrices preserve the arrows here?
​

Undirected graph case

Positions are {x,y}
What matrices leave infinite plane invariant?
{{0,1},{1,0}}
What is ISO(2Z) ?
Rotations + translations
​
In[]:=
Select[Partition[#,2]&/@Tuples[{-1,0,1},4],OrthogonalMatrixQ]
Out[]=
In[]:=
{{1,0},{0,-1}}.{x,y}
Out[]=
{x,-y}
In[]:=
Select[Partition[#,2]&/@Tuples[{-2,-1,0,1,2},4],OrthogonalMatrixQ]
Out[]=

Claim: for the 3-direction case, you have to use complex numbers in the coordinates in order to have a matrix representation

In the general case:

{t,b,x}