When LLMs judge moral dilemmas, do they reach different conclusions in different languages, and if so, why? Two factors could drive such differences: the language of the dilemma itself, or the language in which the model reasons. Standard evaluation conflates these by testing only matched conditions (e.g., English dilemma with English reasoning). We introduce a methodology that separately manipulates each factor, covering also mismatched conditions (e.g., English dilemma with Chinese reasoning), enabling decomposition of their contributions. To study \emph{what} changes, we propose an approach to interpret the moral judgments in terms of Moral Foundations Theory. As a side result, we identify evidence for splitting the Authority dimension into a family-related and an institutional dimension. Applying this methodology to English-Chinese moral judgment with 13 LLMs, we demonstrate its diagnostic power: (1) the framework isolates reasoning-language effects as contributing twice the variance of input-language effects; (2) it detects context-dependency in nearly half of models that standard evaluation misses; and (3) a diagnostic taxonomy translates these patterns into deployment guidance. We release our code and datasets at https://anonymous.4open.science/r/CrossCulturalMoralJudgement.