An often cited statement of Baumert in his book Cyclic difference sets asserts that four well known families of cyclic (4t − 1, 2t − 1, t − 1) difference sets are inequivalent, apart from a small number of exceptions with t 6 8. We are not aware of a proof of this statement in the literature. Three of the families discussed by Baumert have analogous constructions in noncyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases. 2010 Mathematics Subject Classification: 05B20