In SAR image registration, most existing methods consider the image registration as a two-classification problem to construct the pair training samples for training the deep model. However, it is difficult to obtain a mass of given matched-points directly from SAR images as the training samples. Based on this, we propose a multi-class double-transformation network for SAR image registration based on Swin-Transformer. Different from existing methods, the proposed method directly considers each key point as an independent category to construct the multi-classification model for SAR image registration. Then, based on the key points from the reference and sensed images, respectively, a double-transformation network with two branches is designed to search for matched-point pairs. In particular, to weaken the inherent diversity between two SAR images, key points from one image are transformed to the other image, and the transformed image is used as the basic image to capture sub-images corresponding to all key points as the training and testing samples. Moreover, a precise-matching module is designed to increase the reliability of the obtained matched-points by eliminating the inconsistent matched-point pairs given by two branches. Finally, a series of experiments illustrate that the proposed method can achieve higher registration performance compared to existing methods.