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At last, I can see it all. It started when I noticed that on cubic 3d board, a plane including two standard (root 2) diagonals not at right angles (a) automatically includes a third, (b) comprises squares of only one Bishop colour but all four Unicorn colours, and (c) resembles a hex board. The 3d orthogonal and root 3 diagonal are lost but the root 2 diagonal becomes a hex orthogonal, the 2:1:1 (root 6) oblique a hex root 3 diagonal, and the 3:2:1 (root 14) oblique a root 7 hex oblique. Then I realised what the Tetrahedral Chess pattern of 3 diagonals mutually at right angles and 6 orthogonals each at right angles to one other and a diagonal reminded me of. It was the cubic board's orthogonals and diagonals swapped over! Yes, the Tetrahedral board can also be seen as a cubic-cell board with cells of one Bishop colour missing, and the cell notation shown actually fits with this interpretation!