I can think of only two actual coordinate systems for this. One would be to number each of the eight cubes, and designate each face according to which two cubes it connects The other is to assign a binary notatioin, with White's Home Face as 0000 and Black's as 1111. The problem with either of those is that one would likely have to already understand four-dimensional geometry to understand it, and part of the goal here is to help the reader understand four-dimensional geometry. It's not as simple as when using the interior of the tesseract rather than its surface.
The system here starts with the two Faces where White and Black start, designates the four directions (1-4) that one can leave that board, and the two paths (A/B) that can be taken to reach the other side along with four faces (12, 23, 34, 14) that can be used to switch from one path to the other. Except for those last four, all Faces have a designation that describes its relationship to one of the Home Faces, and by implication the other as well. If the designation starts with 0, then you know it's adjacent to White's Home Face; if it starts with 5, then it's adjacent to Black's (I may decide to change those to W and B just to be fully clear). If it ends with A, then you know it's adjacent to AX; if it ends with B, then you know it's adjacent to BX. If its number is 2, then you know it's between 1 and 3; and so forth.
WH (White's Home) and BH (Black's Home) should clearly imply that they are on opposite ends. There are CVs on this site where Black and White don't set up on opposite ends of the board, but they're extremely rare (of the hundred or so that I've perused, I've seen maybe two). When one understands that, then it seems natural that WH is connected to all of the Faces that start with W, while BH is connected to all of the Faces that start with B. This diagram shows how 1, 2, 3, and 4 are arranged for White, and this one for black; I think they'll be even clearer if the sides are designated as W or B instead of 0 or 5. I could also change the middle digits to N/S/E/W; most English-speakers understand the relationships of those.
A and B are arbitrary, and can't be anything else; the only significant thing about either is that it isn't the other one. It's a little like having two hallways connecting two room together, with a door halfway down connecting them (that door representing the "grey" Faces). If I change all the numbers above to letters, then these will probably be called 1 and 2, or perhaps L and H for Low and High.
Edit: I think I will, in fact, make the alphanumeric changes I mention. It's a bit more intuitive.
I can think of only two actual coordinate systems for this. One would be to number each of the eight cubes, and designate each face according to which two cubes it connects The other is to assign a binary notatioin, with White's Home Face as 0000 and Black's as 1111. The problem with either of those is that one would likely have to already understand four-dimensional geometry to understand it, and part of the goal here is to help the reader understand four-dimensional geometry. It's not as simple as when using the interior of the tesseract rather than its surface.
The system here starts with the two Faces where White and Black start, designates the four directions (1-4) that one can leave that board, and the two paths (A/B) that can be taken to reach the other side along with four faces (12, 23, 34, 14) that can be used to switch from one path to the other. Except for those last four, all Faces have a designation that describes its relationship to one of the Home Faces, and by implication the other as well. If the designation starts with 0, then you know it's adjacent to White's Home Face; if it starts with 5, then it's adjacent to Black's (I may decide to change those to W and B just to be fully clear). If it ends with A, then you know it's adjacent to AX; if it ends with B, then you know it's adjacent to BX. If its number is 2, then you know it's between 1 and 3; and so forth.
WH (White's Home) and BH (Black's Home) should clearly imply that they are on opposite ends. There are CVs on this site where Black and White don't set up on opposite ends of the board, but they're extremely rare (of the hundred or so that I've perused, I've seen maybe two). When one understands that, then it seems natural that WH is connected to all of the Faces that start with W, while BH is connected to all of the Faces that start with B. This diagram shows how 1, 2, 3, and 4 are arranged for White, and this one for black; I think they'll be even clearer if the sides are designated as W or B instead of 0 or 5. I could also change the middle digits to N/S/E/W; most English-speakers understand the relationships of those.
A and B are arbitrary, and can't be anything else; the only significant thing about either is that it isn't the other one. It's a little like having two hallways connecting two room together, with a door halfway down connecting them (that door representing the "grey" Faces). If I change all the numbers above to letters, then these will probably be called 1 and 2, or perhaps L and H for Low and High.
Edit: I think I will, in fact, make the alphanumeric changes I mention. It's a bit more intuitive.