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A new general paradigm of dynamic-range-preserving one-to-one mapping-infinity-norm rotations, analogous to the general 2-norm rotations, are proposed in this paper. Analogous to the well-known discrete cosine transforms, the linear 2-norm rotation transforms which preserve the 2-norm of the rotated vectors, the proposed infinity-norm rotation transforms are piecewise linear transforms which preserve the infinity-norm of vectors. Besides the advantages of perfect reversibility, in-place calculation and dynamic range preservation, the infinity-norm rotation transforms also have good energy-compact ability, which is suitable for signal compression and analysis. It can be implemented by shear transforms based on the 2-D rotation factorization of similar orthogonal transform matrices, such as DCT matrices. The performance of the new transforms is illustrated with 2-D patterns and histograms. Its good performance in lossy and lossless image compression, compared with other integer reversible transforms, is demonstrated in the experiments.