While the inconsistency of LLMs is not a novel topic, prior research has predominantly addressed two types of generative inconsistencies: i) Randomness Inconsistency: running the same LLM multiple trials, yielding varying responses; ii) Paraphrase Inconsistency: paraphrased prompts result in different responses from the same LLM. Randomness Inconsistency arises from the inherent randomness due to stochastic sampling in generative models, while Paraphrase Inconsistency is a consequence of the language modeling objectives, where paraphrased prompts alter the distribution of vocabulary logits. This research discovers Prompt-Reverse Inconsistency (PRIN), a new form of LLM self-inconsistency: given a question and a couple of LLM-generated answer candidates, the LLM often has conflicting responses when prompted "Which are correct answers?" and "Which are incorrect answers?". PRIN poses a big concern as it undermines the credibility of LLM-as-a-judge, and suggests a challenge for LLMs to adhere to basic logical rules. We conduct a series of experiments to investigate PRIN, examining the extent of PRIN across different LLMs, methods to mitigate it, potential applications, and its relationship with Randomness Inconsistency and Paraphrase Inconsistency. As the first study to explore PRIN, our findings offer valuable insights into the inner workings of LLMs and contribute to advancing trustworthy AI.