Signal degradation is ubiquitous, and computational restoration of degraded signal has been investigated for many years. Recently, it is reported that the capability of signal restoration is fundamentally limited by the so-called perception-distortion tradeoff, i.e. the distortion and the perceptual difference between the restored signal and the ideal "original" signal cannot be made both minimal simultaneously. Distortion corresponds to signal fidelity and perceptual difference corresponds to perceptual naturalness, both of which are important metrics in practice. Besides, there is another dimension worthy of consideration--the semantic quality of the restored signal, i.e. the utility of the signal for recognition purpose. In this paper, we extend the previous perception-distortion tradeoff to the case of classification-distortion-perception (CDP) tradeoff, where we introduced the classification error rate of the restored signal in addition to distortion and perceptual difference. In particular, we consider the classification error rate achieved on the restored signal using a predefined classifier as a representative metric for semantic quality. We rigorously prove the existence of the CDP tradeoff, i.e. the distortion, perceptual difference, and classification error rate cannot be made all minimal simultaneously. We also provide both simulation and experimental results to showcase the CDP tradeoff. Our findings can be useful especially for computer vision research where some low-level vision tasks (signal restoration) serve for high-level vision tasks (visual understanding). Our code and models have been published.