The appropriate characterization of the test material, used for subjective evaluation tests and for benchmarking image and video processing algorithms and quality metrics, can be crucial in order to perform comparative studies that provide useful insights. This paper focuses on the characterisation of 360-degree images. We discuss why it is important to take into account the geometry of the signal and the interactive nature of 360-degree content navigation, for a perceptual characterization of these signals. Particularly, we show that the computation of classical indicators of spatial complexity, commonly used for 2D images, might lead to different conclusions depending on the geometrical domain used to represent the 360-degree signal. Finally, new complexity measures based on the analysis of visual attention and content exploration patterns are proposed.