The Dirac equation is shown to admit a natural algebra P of global pseudodifferential operators, characterised by the property that the 'Heisenberg representation' \(A\to \exp (iHt)A \exp (-iHt)=A_ t\) leaves P invariant. Certain modifications of most standard observables are required if the algebra P is to be interpreted as an algebra of observables. An Egorov type theorem can be applied to gain direct insight into particle orbits spin propagation not available with other algebras. Some of the inconsistencies of classical theory naturally disappear when this point of view is adopted.