Two vectors in Z3 are called 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 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. © 2011 Elsevier Inc.