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Hyperspectral (HS) images are characterized by approximately contiguous spectral information, enabling the fine identification of materials by capturing subtle spectral discrepancies. Owing to their excellent locally contextual modeling ability, convolutional neural networks (CNNs) have been proven to be a powerful feature extractor in HS image classification. However, CNNs fail to mine and represent the sequence attributes of spectral signatures well due to the limitations of their inherent network backbone. To solve this issue, we rethink HS image classification from a sequential perspective with transformers, and propose a novel backbone network called \ul{SpectralFormer}. Beyond band-wise representations in classic transformers, SpectralFormer is capable of learning spectrally local sequence information from neighboring bands of HS images, yielding group-wise spectral embeddings. More significantly, to reduce the possibility of losing valuable information in the layer-wise propagation process, we devise a cross-layer skip connection to convey memory-like components from shallow to deep layers by adaptively learning to fuse "soft" residuals across layers. It is worth noting that the proposed SpectralFormer is a highly flexible backbone network, which can be applicable to both pixel- and patch-wise inputs. We evaluate the classification performance of the proposed SpectralFormer on three HS datasets by conducting extensive experiments, showing the superiority over classic transformers and achieving a significant improvement in comparison with state-of-the-art backbone networks. The codes of this work will be available at https://github.com/danfenghong/IEEE_TGRS_SpectralFormer for the sake of reproducibility.