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Remote sensing (RS) image scene classification is an important research topic in the RS community, which aims to assign the semantics to the land covers. Recently, due to the strong behavior of convolutional neural network (CNN) in feature representation, the growing number of CNN-based classification methods has been proposed for RS images. Although they achieve cracking performance, there is still some room for improvement. First, apart from the global information, the local features are crucial to distinguish the RS images. The existing networks are good at capturing the global features since the CNNs' hierarchical structure and the nonlinear fitting capacity. However, the local features are not always emphasized. Second, to obtain satisfactory classification results, the distances of RS images from the same/different classes should be minimized/maximized. Nevertheless, these key points in pattern classification do not get the attention they deserve. To overcome the limitation mentioned above, we propose a new CNN named attention consistent network (ACNet) based on the Siamese network in this article. First, due to the dual-branch structure of ACNet, the input data are the image pairs that are obtained by the spatial rotation. This helps our model to fully explore the global features from RS images. Second, we introduce different attention techniques to mine the objects' information from RS images comprehensively. Third, considering the influence of the spatial rotation and the similarities between RS images, we develop an attention consistent model to unify the salient regions and impact/separate the RS images from the same/different semantic categories. Finally, the classification results can be obtained using the learned features. Three popular RS scene datasets are selected to validate our ACNet. Compared with some existing networks, the proposed method can achieve better performance. The encouraging results illustrate that ACNet is effective for the RS image scene classification. The source codes of this method can be found in https://github.com/TangXu-Group/Remote-Sensing-Images-Classification/tree/main/GLCnet.