This paper contains two well differentiated parts. The first part contains a revision of the axioms defining a hypergroup. In the classical definition of hypergroups (often called DJS-hypergroups after Dunkl, Jewett and Spector), the set \(\mathfrak{C}(K)\) of all compact subsets of a locally compact space \(X\) is endowed with the Michael topology and the mapping from \(K\times K\) into \(\mathfrak{C}(K)\) defined by \((x,y)\mapsto \text{supp}(\delta_x * \delta_y)\), is required to be continuous (see, for instance, \textit{W. R. Bloom} and \textit{H. Heyer} [Harmonic analysis of probability measures on hypergroups. De Gruyter Studies in Math. 20 (Berlin 1995; Zbl 0828.43005)]). In this new axiomatic, topologies on the space of compact subsets are not used and the verification that a particular measure algebra is a hypergroup becomes easier. It should be noted that the class of hypergroups thus obtained turns out to be slightly larger than the usual class of DJS-hypergroups. The authors devote most of this part of the paper to prove, within this new axiomatic, some of the classical theorems of harmonic analysis, like Bochner's or Plancherel's. In the second part of the paper the authors attempt a classification of compact commutative hypergroups based on the spectral properties of compact commutative hypergroup measure algebras. In particular, symmetry and idempotents of measure algebras and the existence of Sidon sets in duals of hypergroups are investigated. The main tools introduced here are the Dixmier symbolic calculus, leading to an external construction of polynomial bounded approximate identities, and the maximum subgroup which provides a way to relate harmonic analysis on compact hypergroups to harmonic analysis on compact groups. Up to six concrete examples are presented to illustrate how the maximum subgroup can be placed in a compact commutative hypergroup.