Unified image fusion aims to integrate complementary information from multi-source images, enhancing image quality through a unified framework applicable to diverse fusion tasks. While treating all fusion tasks as a unified problem facilitates task-invariant knowledge sharing, it often overlooks task-specific characteristics, thereby limiting the overall performance. Existing general image fusion methods incorporate explicit task identification to enable adaptation to different fusion tasks. However, this dependence during inference restricts the model's generalization to unseen fusion tasks. To address these issues, we propose a novel unified image fusion framework named "TITA", which dynamically balances both Task-invariant Interaction and Task-specific Adaptation. For task-invariant interaction, we introduce the Interaction-enhanced Pixel Attention (IPA) module to enhance pixel-wise interactions for better multi-source complementary information extraction. For task-specific adaptation, the Operation-based Adaptive Fusion (OAF) module dynamically adjusts operation weights based on task properties. Additionally, we incorporate the Fast Adaptive Multitask Optimization (FAMO) strategy to mitigate the impact of gradient conflicts across tasks during joint training. Extensive experiments demonstrate that TITA not only achieves competitive performance compared to specialized methods across three image fusion scenarios but also exhibits strong generalization to unseen fusion tasks.