The rapid and accurate damage assessment of buildings plays a critical role in disaster response. Based on pairs of pre- and post-disaster remote sensing images, effective building damage level assessment can be conducted. However, most existing methods are based on Convolutional Neural Network, which has limited ability to learn the global context. An attention mechanism helps ameliorate this problem. Hierarchical Transformer has powerful potential in the remote sensing field with strong global modeling capability. In this paper, we propose a novel two-stage damage assessment framework called SDAFormer, which embeds a symmetric hierarchical Transformer into a siamese U-Net-like network. In the first stage, the pre-disaster image is fed into a segmentation network for building localization. In the second stage, a two-branch damage classification network is established based on weights shared from the first stage. Then, pre- and post-disaster images are delivered to the network separately for damage assessment. Moreover, a spatial fusion module is designed to improve feature representation capability by building pixel-level correlation, which establishes spatial information in Swin Transformer blocks. The proposed framework achieves significant improvement on the large-scale building damage assessment dataset—xBD.