Distribution-preserving integrated sensing and communication is investigated in this paper. In addition to the distortion constraint, we impose another constraint on the distance between the reconstructed sequence distribution and the original state distribution to force the system to preserve the statistical property of the channel states. An inner bound of the distribution-preserving capacity-distortion region is provided with some capacity region results under special cases. Furthermore, we consider the case where the system aims to keep the reconstructed sequence secret from an eavesdropper who also observes the channel output and receives rate-limited side information about the estimator. An inner bound of the tradeoff region and a capacity-achieving special case are presented. In addition, we provide some numerical examples to illustrate the tradeoff between the communication rate, distortion, and the preservation of the distribution.