Moral self-correction has emerged as a promising approach for aligning the output of Large Language Models (LLMs) with human moral values. However, moral self-correction techniques are subject to two primary paradoxes. First, despite empirical and theoretical evidence to support the effectiveness of self-correction, this LLM capability only operates at a superficial level. Second, while LLMs possess the capability of self-diagnosing immoral aspects of their output, they struggle to identify the cause of this moral inconsistency during their self-correction process. To better understand and address these paradoxes, we analyze the discourse constructions in fine-tuning corpora designed to enhance moral self-correction, uncovering the existence of the heuristics underlying effective constructions. We demonstrate that moral self-correction relies on discourse constructions that reflect heuristic shortcuts, and that the presence of these heuristic shortcuts during self-correction leads to inconsistency when attempting to enhance both self-correction and self-diagnosis capabilities jointly. Based on our findings, we propose a solution to improve moral self-correction by leveraging the heuristics of curated datasets. We also highlight the generalization challenges of this capability, particularly in terms of learning from situated context and model scales.