Many implicit inferences exist in text depending on how it is structured that can critically impact the text's interpretation and meaning. One such structural aspect present in text with chronology is the order of its presentation. For narratives or stories, this is known as the narrative order. Reordering a narrative can impact the temporal, causal, event-based, and other inferences readers draw from it, which in turn can have strong effects both on its interpretation and interestingness. In this paper, we propose and investigate the task of Narrative Reordering (NAREOR) which involves rewriting a given story in a different narrative order while preserving its plot. We present a dataset, NAREORC, with human rewritings of stories within ROCStories in non-linear orders, and conduct a detailed analysis of it. Further, we propose novel task-specific training methods with suitable evaluation metrics. We perform experiments on NAREORC using state-of-the-art models such as BART and T5 and conduct extensive automatic and human evaluations. We demonstrate that although our models can perform decently, NAREOR is a challenging task with potential for further exploration. We also investigate two applications of NAREOR: generation of more interesting variations of stories and serving as adversarial sets for temporal/event-related tasks, besides discussing other prospective ones, such as for pedagogical setups related to language skills like essay writing and applications to medicine involving clinical narratives.