Understanding human-object interaction is a fundamental challenge in computer vision and robotics. Crucial to it is the ability to infer "object affordances" from visual data, namely the types of interaction supported by an object of interest and the object parts involved. Such inference can be approached as an "affordance reasoning" task, where object affordances are recognized and localized as image heatmaps, and as an "affordance segmentation" task, where affordance labels are obtained at a more detailed, image pixel level. To tackle the two tasks, existing methods typically: (i) treat them independently; (ii) adopt static image-based models, ignoring the temporal aspect of human-object interaction; and / or (iii) require additional strong supervision concerning object class and location. In this paper, we focus on both tasks, while addressing all three aforementioned shortcomings. For this purpose, we propose a deep-learning based dual encoder-decoder model for joint affordance reasoning and segmentation, which learns from our recently introduced SOR3D-AFF corpus of RGB-D human-object interaction videos, without relying on object localization and classification. The basic components of the model comprise: (i) two parallel encoders that capture spatio-temporal interaction information; (ii) a reasoning decoder that predicts affordance heatmaps, assisted by an affordance classifier and an attention mechanism; and (iii) a segmentation decoder that exploits the predicted heatmap to yield pixel-level affordance segmentation. All modules are jointly trained, while the system can operate on both static images and videos. The approach is evaluated on four datasets, surpassing the current state-of-the-art in both affordance reasoning and segmentation.