The rise of multimodal content on social platforms has led to the rapid spread of complex and persuasive false narratives, combining of text and images. Traditional rumor detection models attempt to identify such content by relying on textual cues or employing shallow multimodal fusion techniques. However, these methods often assume a simplistic one-to-one alignment between modalities, overlooking the richer hierarchical relationships across modalities, failing to capture the layered structure of meaning. In this paper, we present RumorCone, a novel method that employs hyperbolic geometry in order to preserve hierarchical, non-linear relationships, rather than representing them at a flat semantic level. First, RumorCone decomposes image and text content into three levels: base, mid, and high-level abstractions, and embeds them in hyperbolic space to model their tree-like semantic structure. Second, a dynamic hyperbolic multimodal attention mechanism aligns features across modalities and levels, and a flexible fusion strategy adjusts the contribution of each modality based on alignment quality. Our experiments indicate the importance of hierarchical semantic modeling for robust and interpretable multimodal rumor detection.