As far as I can tell this question has never been asked before. There are several questions that touch on related issues, but as far as I can see none of them have provided a definitive answer to this question. Furthermore, in at least one case the most upvoted and second most upvoted answer have implicitly disagreed with each other.
Textbooks often caution against interpreting main effects in Two-Way ANOVA when the interaction effect is significant. Here's a (mild) example of this, from pp562-563 of Hatcher's "Step-by-step Basic Statistics Using SAS: Student Guide":
When you perform a two-way ANOVA, it is possible that you will find that (a) the interaction term is statistically significant, and (b) one or both of the main effects are also statistically significant. When you prepare a report summarizing the results, you will certainly discuss nature of your significant interaction. But is it also acceptable to discuss and interpret the main effects that were significant?
There is some disagreement between statisticians in answering this question. Some statisticians argue that, if the interaction is significant, you should not interpret the main effects at all, even if they are significant. Others take a less extreme approach. They say that it is acceptable to interpret significant main effects, as long as long as the primary interpretation of the results focuses on the interaction (if it is significant).
Do I assume correctly that the hard-line stance ("you should not interpret the main effects at all") is incorrect, but that the interpretation must nonetheless change if the interaction effect is significant? If so, how does the interpretation change?