Most previous work on gloss perception has examined the strength and sharpness of specular reflections in simple bidirectional reflectance distribution functions (BRDFs) having a single specular component. However, BRDFs can be substantially more complex and it is interesting to ask how many additional perceptual dimensions there could be in the visual representation of surface reflectance qualities. To address this, we tested materials with two specular components that elicit an impression of hazy gloss. Stimuli were renderings of irregularly shaped objects under environment illumination, with either a single Ward specular BRDF component (Ward, 1992), or two such components, with the same total specular reflectance but different sharpness parameters, yielding both sharp and blurry highlights simultaneously. Differently shaped objects were presented side by side in matching, discrimination, and rating tasks. Our results show that observers mainly attend to the sharpest reflections in matching tasks, but they can indeed discriminate between single-component and two-component specular materials in discrimination and rating tasks. The results reveal an additional perceptual dimension of gloss-beyond strength and sharpness-akin to ''haze gloss'' (Hunter & Harold, 1987). However, neither the physical measurements of Hunter and Harold nor the kurtosis of the specular term predict perception in our tasks. We suggest the visual system may use a decomposition of specular reflections in the perception of hazy gloss, and we compare two possible candidates: a physical representation made of two gloss components, and an alternative representation made of a central gloss component and a surrounding halo component.