(Joint work with Robert T. Powers). The group of local unitary cocycles of an E0-semigroup, also called the gauge group, acts in a natural way on the set of units of the E0-semigroup. It follows from work of William Arveson that for a completely spatial E0-semigroup of index n this action is not only transitive, but also n+ 1-fold transitive: the group acts transitively on the set of normalized n + 1-tuples of independent units. In this talk we will analyze the local unitary cocycles of a class of spatial E0-semigroups which we call strongly spatial. As an application we will present examples of strongly spatial E0-semigroups of index 1 whose gauge groups are not 2-fold transitive.