Three-way three-mode data are collected regularly in scientific research and yield information on the relation between three sets of entities. To summarize the information in such data, three-way component methods like CANDECOMP/PARAFAC (CP) and Tucker3 are often used. When applying CP and Tucker3 in empirical practice, one should be cautious, however, because they rely on very strict assumptions. We argue that imposing these assumptions may obscure interesting structural information included in the data and may lead to substantive conclusions that are appropriate for some part of the data only. As a way out, this paper demonstrates that this structural information may be elegantly captured by means of component methods for multiset data, that is to say, simultaneous component analysis (SCA) and its clusterwise extension (clusterwise SCA).