Three novel chalcogen-bonded cocrystals featuring 3,4-dicyano-1,2,5-selenodiazole (C4N4Se) or 3,4-dicyano-1,2,5-tellurodiazole (C4N4Te) as chalcogenbond donors and hydroquinone (C6H6O2), tetraphenylphosphonium chloride (C24H20P+ center dot Cl-) or tetraethylphosphonium chloride (C8H20P+ center dot Cl-) as chalcogen-bond acceptors have been prepared and characterized by single-crystal X-ray diffraction (XRD), powder X-ray diffraction and Se-77/Te-125 magic-angle spinning solid-state NMR spectroscopy. The single-crystal XRD results show that the chalcogenodiazole molecules interact with the electron donors through two sigma -holes on each of the chalcogen atoms, which results in highly directional and moderately strong chalcogen bonds. Powder XRD confirms that the crystalline phases are preserved upon moderate grinding of the samples for solid-state NMR experiments. Measurement of Se-77 and Te-125 chemical shift tensors via magic-angle spinning solid-state NMR spectroscopy confirms the number of magnetically unique chalcogen sites in each asymmetric unit and reveals the impact of chalcogen-bond formation on the local electronic structure. These NMR data are further assessed in the context of analogous data for a wider range of crystalline chalcogen-bonded systems.