 Orthogonality of hypercube body diagonals

Two sign-vector vertices of a centered hypercube differing in $h$ coordinates have dot product $n-2h$. Thus their body diagonals can be orthogonal only in even dimension. A collection of mutually orthogonal nonzero diagonals has at most $n$ members; achieving $n$ is equivalent to the existence of a Hadamard matrix of order $n$. For unoriented lines the angle is determined by $|n-2h|/n$, while oriented vertex vectors retain the sign of their dot product.