Texture classification plays an important role for various computer vision tasks. Depending upon the powerful feature extraction capability, convolutional neural network (CNN)-based texture classification methods have attracted extensive attention. However, there still exist many challenges, such as the extraction of multilevel texture features and the exploration of multidirectional relationships. To address the problem, this paper proposes the compressed wavelet tensor attention capsule network (CWTACapsNet), which integrates multiscale wavelet decomposition, tensor attention blocks, and quantization techniques into the framework of capsule neural network. Specifically, the multilevel wavelet decomposition is in charge of extracting multiscale spectral features in frequency domain; in addition, the tensor attention blocks explore the multidimensional dependencies of convolutional feature channels, and the quantization techniques make the computational storage complexities be suitable for edge computing requirements. The proposed CWTACapsNet provides an efficient way to explore spatial domain features, frequency domain features, and their dependencies which are useful for most texture classification tasks. Furthermore, CWTACapsNet benefits from quantization techniques and is suitable for edge computing applications. Experimental results on several texture datasets show that the proposed CWTACapsNet outperforms the state-of-the-art texture classification methods not only in accuracy but also in robustness.