Understanding the spatial arrangements of atom-centered coordination octahedra is crucial for relating structures to properties for many materials families. Traditional case-by-case inspection becomes a prohibitive task for discovering trends and similarities in large datasets. Here, we operationalize chemical intuition to automate the geometric parsing, quantification, and classification of coordination octahedral networks using unsupervised machine learning. We apply the workflow to analyze two datasets to demonstrate its effectiveness. For computationally generated single oxide perovskite (ABO$_{3}$) polymorphs, we uncover axis-dependent tilting trends which assist in detecting oxidation state changes. For hybrid iodoplumbates (A$_x$Pb$_y$I$_z$) from measured structures, we taxonomize their octahedral networks, revealing a Pauling-like connectivity rule for the coordination environment and the design principles underpinning their structural diversity. Our results offer a glimpse into the vast design space of atomic octahedral networks in materials chemistry and inform high-throughput, targeted screening of specific structure types.