The problem of cluster analysis eludes a unique mathematical definition. Instead, a variety of different instantiations of the problem can be defined using specific measures of internal cluster validity. In turn, such internal cluster validity measures rely on quantifying dissimilarity between entities. This article explores the interaction between dissimilarity measures and internal cluster validity techniques in the context of multi-objective clustering. It does so by contrasting two conceptually different approaches to multi-objective clustering: the multi-criterion clustering algorithm Δ-MOCK, designed to optimise different measures of internal cluster validity over a single dissimilarity space, and the multi-view clustering algorithm MVMC, designed to optimise a single measure of internal cluster validity over distinct dissimilarity spaces. Our comparison highlights the interchangeable roles of distance functions and measures of internal cluster validity, which paves the way for the future design of a flexible, dual-purpose approach to multi-objective clustering.