Author’s summary: Several equivalent definitions of anticommutativity for self-adjoint operators are presented. It is shown that the anticommutativity of two self-adjoint operators A and B, satisfying $$<Af| Bg>+<Bf| Ab>=0$$ for all f and g in the intersection of the domains of A and B, is equivalent to the self-adjointness of their sum and difference. In studying the structure of anticommuting self-adjoint operators A and B, attention may be restricted to the case where both A and B are injective. In that case, A is unitarily equivalent to -A. Moreover, A and B are uniquely determined by a triplet (b,P,Q), where b is unitary, and P and Q are commuting, injective, positive, and self- adjoint operators. This is a natural correspondence.