A cohomology operation encoding the octahedral axiom

We introduce a new secondary cohomology operation, the naturality defect product, associated with a graded natural transformation between the cohomology functors of two differential graded functors, and study its main properties. We then prove that a pretriangulated category satisfies the octahedral axiom if and only if the naturality defect product associated with the natural isomorphism given by a theorem of Heller vanishes on the homotopy category of acyclic complexes over it. This characterization extends immediately to n-angulated categories. As an application, we exhibit non-vanishing instances for the recently found examples of pretriangulated categories which are not triangulated. Finally, we show that horizontal composition of these natural isomorphisms preserves n-angulated structures, and we use it to obtain, from each of these examples and for every r >= 1, a pre-3r-angulated category which is not 3r-angulated.