Symmetries and symmetry-breaking play significant roles in data security. While remote sensing images, being extremely sensitive geospatial data, require protection against tampering or destruction, as well as assurance of the reliability of the data source during application. In view of the increasing complexity of data security of remote sensing images, a single watermark algorithm is no longer adequate to meet the demand of sophisticated applications. Therefore, this study proposes a dual watermarking algorithm that considers both integrity authentication and copyright protection of remote sensing images. The algorithm utilizes Discrete Wavelet Transform (DWT) to decompose remote sensing images, then constructs integrity watermark information by applying Chebyshev mapping to the mean of horizontal and vertical components. This semi-fragile watermark information is embedded into the high-frequency region of DWT using Quantization Index Modulation (QIM). On the other hand, the robust watermarking uses entropy to determine the embedding position within the DWT domain. It combines the stability of Singular Value Decomposition (SVD) and embeds the watermark according to the relationship between the singular values of horizontal, vertical, and high-frequency components. The experiment showed that the proposed watermarking successfully maintains a high level of invisibility even if embedded with dual watermarks. The semi-fragile watermark can accurately identify tampered regions in remote sensing images under conventional image processing. Moreover, the robust watermark exhibits excellent resistance to various attacks such as noise, filtering, compression, panning, rotating, and scaling.