It’s a question I get pretty often, and it’s a more straightforward answer than most.
There is really only one situation possible in which an interaction is significant, but the main effects are not: a cross-over interaction.
Unlike many terms in statistics, a cross-over interaction is exactly what it says: the means cross over each other in the different situations. Here’s an example of a two-by-two ANOVA with a cross-over interaction:
The two grey dots indicate the main effect means for Factor A. Their height is pretty much the same, so there would be no main effect for Factor A.
The two grey Xs indicate the main effect means for Factor B. Sure, the B1 mean is slightly higher than the B2 mean, but not by much. In most data sets, this difference would not be significant.
But there clearly is an interaction. The difference in the B1 means is clearly different at A1 than it is at A2 (one difference is positive, the other negative).
So yes, you would would interpret this interaction and it is giving you meaningful information.
What does it mean? You’d say there is no overall effect of either Factor A or Factor B, but there is a crossover interaction. The effect of B on the dependent variable is opposite, depending on the value of Factor A.