In a rapidly growing field of model training there is a constant practical interest in parameter-efficient fine-tuning and various techniques that use a small amount of training data to adapt the model to a narrow task. However, there is an open question: how to combine several adapters tuned for different tasks into one which is able to yield adequate results on both tasks? Specifically, merging subject and style adapters for generative models remains unresolved. In this paper we seek to show that in the case of orthogonal fine-tuning (OFT), we can use structured orthogonal parametrization and its geometric properties to get the formulas for training-free adapter merging. In particular, we derive the structure of the manifold formed by the recently proposed Group-and-Shuffle ($\mathcal{GS}$) orthogonal matrices, and obtain efficient formulas for the geodesics approximation between two points. Additionally, we propose a $\text{spectra restoration}$ transform that restores spectral properties of the merged adapter for higher-quality fusion. We conduct experiments in subject-driven generation tasks showing that our technique to merge two $\mathcal{GS}$ orthogonal matrices is capable of uniting concept and style features of different adapters. To the best of our knowledge, this is the first training-free method for merging multiplicative orthogonal adapters. Code is available via the $\href{https://github.com/ControlGenAI/OrthoFuse}{link}$.