ABSTRACT During its strong outburst of 2015 June/July, the X-ray transient V404 Cygni (=GS2023+338) was observed up to a level of 50 Crab in the hard X-ray domain. We focus here on a particularly intense episode preceeding a definitive decline of the source activity. We benefit from large signal-to-noise ratios to investigate the source spectral variability, on a timescale of five minutes. A hardness–intensity study of three broad bands reveals clearly different behaviors at low and high energy (below and above ∼100 keV). In particular, on two occasions, the source intensity varies by a factor of 3–4 in amplitude while keeping the same spectral shape. On the other hand, at the end of the major flare, the emission presents a clear anticorrelation between flux and hardness. These behaviors strongly suggest the presence of two spectral components related to emission processes varying in a largely independent way. The first component ( E < 100–150 keV) is classically identified with a Comptonizing thermal electron population, and requires either an unusual seed photon population or a specific geometry with strong absorbing/reflecting material. The second component is modeled by a cutoff power-law, which could correspond to a second hotter Comptonizing population or another mechanism (synchrotron, non-thermal Comptonization...). In the framework of such a model, hardness–intensity and flux–flux diagrams clearly demonstrate that the source evolution follows a well-organized underlying scheme. They reveal unique information about the hard X-ray emission processes and connections between them.