Sorry for the absence. Took a nasty fall that made typing impossible so, have been recouping. I'm better enough
I think this level of question needs to move away from the "tool" (Neo) and into the math or graph theory. Love the question about "who owns" the "direction" whose answer is "both" and/or neither. Let me explain.
A "graph" is visual to humans starting with the now famous bridge problem. But the math of graph theory is not a visual rather than a model and a means to ask questions of the nodes and relationships. The best way to see how this looks for the "ownership" understanding, is to see a graph as a matrix.
Let's keep it super simple and imagine a 2 node graph. In this case, we can add a relationship...but remember, a graph G can have nodes and NO relationships. So I can't inset a table it seems but imagine my matrix[a,b] with cells [a,a],[a,b],[b,a],[b,b]. These are the four "points of view" of the two nodes. I saw POV because 'a' can look at itself and at 'b'. And 'b' can look at itself and at 'a'. NOTE: while 'a' and 'b' can ONLY see themselves from a single POV, they can see each other from two perspectives a->b and b->a. I'm not addressing direction yet, just POV. Simple graph theory is rather basic but this elegance of design enables very complex ideas. But for now, let's stick to the basics.
Assuming (arbitrary rule I'm saying here because it's consistent with how Neo is designed but NOT how graph theory limits the math) that any POV in our graph G can have either a "0" or a "1" as the value. The matrix then, can range from all zeros to all ones. Each cell that has a "1" in it moves from the column label to the row label (you could do either direction, just be consistent). And that "1" is THE DIRECTION.
So you can see how ownership of the direction is intimate with the matrix that represents it. This also addresses how you can NOT have a relationship to a relationship (i.e., when limited by this design concept of using a matrix.
So, hopefully this is helpful. I like how you are exploring graph issues like a graph's dimensionality. I do know of tons of work in graph DB design around adding to this most elegant matrix foundation. But most of these efforts, IMHO, are lazy and miss out on what I think is more the "next step". Instead of adding dimensions to the model, I'd say we should spend way more time addressing matrix multiplication, dot product, vector involvement, etc. This you can see in the graph algorithm work being done at Neo and other places in the graphista community. This is a huge need in the ML and AI work where graphs are going to become so key. Once we have that in place and performant, then the ideas of exploring multi-dimensions beyond what all that will uncover, will be useful to explore (if needed which I debate BTW).
Thanks and HTH.