With the increasing prevalence of cloud computing platforms, ensuring data privacy during the cloud-based image-related services such as classification has become crucial. In this study, we propose a novel privacy-preserving image classification scheme that enables the direct application of classifiers trained in the plaintext domain to classify encrypted images without the need of retraining a dedicated classifier. Moreover, encrypted images can be decrypted back into their original form with high fidelity (recoverable) using a secret key. Specifically, our proposed scheme involves utilizing a feature extractor and an encoder to mask the plaintext image through a newly designed Noise-like Adversarial Example (NAE). Such an NAE not only introduces a noise-like visual appearance to the encrypted image but also compels the target classifier to predict the ciphertext as the same label as the original plaintext image. At the decoding phase, we adopt a Symmetric Residual Learning (SRL) framework for restoring the plaintext image with minimal degradation. Extensive experiments demonstrate that (1) the classification accuracy of the classifier trained in the plaintext domain remains the same in both the ciphertext and plaintext domains; (2) the encrypted images can be recovered into their original form with an average PSNR of up to 51+ dB for the SVHN dataset and 48+ dB for the VGGFace2 dataset; (3) our system exhibits satisfactory generalization capability on the encryption, decryption, and classification tasks across datasets that are different from the training one; and (4) a high-level of security is achieved against three potential threat models. The code is available at https://github.com/csjunjun/RIC.git.