Deep learning has become increasingly important in remote sensing image classification due to its ability to extract semantic information from complex data. Classification tasks often include predefined label hierarchies that represent the semantic relationships among classes. However, these hierarchies are frequently overlooked, and most approaches focus only on fine-grained classification schemes. In this paper, we present a novel Semantics-Aware Hierarchical Consensus (SAHC) approach to learn hierarchical features and relationships by integrating hierarchy-specific classification heads within a deep network architecture, each specialized in different degrees of class granularity. The proposed approach employs trainable hierarchy matrices, which guide the network through the learning of the hierarchical structure in a self-consistent manner. Furthermore, we introduce a hierarchical consensus mechanism to ensure aligned probability distributions across different hierarchical levels. This mechanism acts as a weighted ensemble being able to effectively leverage the inherent structure of the hierarchical classification task. The proposed SAHC method is evaluated on two benchmark datasets with different degrees of hierarchical complexity on different tasks, considering varying spectral and spatial resolutions. Experimental results show both the effectiveness of the proposed approach in guiding network learning and the robustness of the hierarchical consensus for remote sensing image classification tasks. The codes will be released at https://github.com/rslab-unitrento/sahc.