Hyperspectral image (HSI) restoration is a fundamental challenge in computational imaging and computer vision. It involves ill-posed inverse problems, such as inpainting and super-resolution. Although deep learning methods have transformed the field through data-driven learning, their effectiveness hinges on access to meticulously curated ground-truth datasets. This fundamentally restricts their applicability in real-world scenarios where such data is unavailable. This paper presents SHARE (Single Hyperspectral Image Restoration with Equivariance), a fully unsupervised framework that unifies geometric equivariance principles with low-rank spectral modelling to eliminate the need for ground truth. SHARE's core concept is to exploit the intrinsic invariance of hyperspectral structures under differentiable geometric transformations (e.g. rotations and scaling) to derive self-supervision signals through equivariance consistency constraints. Our novel Dynamic Adaptive Spectral Attention (DASA) module further enhances this paradigm shift by explicitly encoding the global low-rank property of HSI and adaptively refining local spectral-spatial correlations through learnable attention mechanisms. Extensive experiments on HSI inpainting and super-resolution tasks demonstrate the effectiveness of SHARE. Our method outperforms many state-of-the-art unsupervised approaches and achieves performance comparable to that of supervised methods. We hope that our approach will shed new light on HSI restoration and broader scientific imaging scenarios. The code will be released at https://github.com/xuwayyy/SHARE.