  Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and
have the same length. The paper describes twin pairs using cubic lattices, and
counts the number of twin pairs with a given length. Integers $M$ with the
property that each integral vector with length $\sqrt{M}$ has a twin are called
twin-complete. They are completely characterized modulo a famous conjecture in
number theory. The main tool is the decomposition theory of Hurwitz integral
quaternions. Throughout the paper we made a concerted effort to keep the
exposition as elementary as possible.
