Acquiring commonsense knowledge and reasoning is an important goal in modern NLP research. Despite much progress, there is still a lack of understanding (especially at scale) of the nature of commonsense knowledge itself. A potential source of structured commonsense knowledge that could be used to derive insights is ConceptNet. In particular, ConceptNet contains several coarse-grained relations, including HasContext, FormOf and SymbolOf, which can prove invaluable in understanding broad, but critically important, commonsense notions such as 'context'. In this article, we present a methodology based on unsupervised knowledge graph representation learning and clustering to reveal and study substructures in three heavily used commonsense relations in ConceptNet. Our results show that, despite having an 'official' definition in ConceptNet, many of these commonsense relations exhibit considerable sub-structure. In the future, therefore, such relations could be sub-divided into other relations with more refined definitions. We also supplement our core study with visualizations and qualitative analyses.