When interpreting interactions in a factorial ANOVA, is it necessary to look at the residual cell means? In their 1989 paper "Definition and Interpretation of Interaction Effects", Rosnow & Rosenthal write:

When interaction is claimed in a factorial arrangement, the results
  almost always require more detailed analysis than is typically
  reported in our primary journals. In reporting interactions, research
  psychologists have gotten into the habit of examining only the
  differences between the original cell means (the simple effects)
  instead of more properly examining the residuals, or leftover effects,
  after the lower order effects have been removed.

And then later in the same article:

The point of this article is to emphasize that if investigators are 
  claiming to speak of an interaction, the exercise of looking at the
  corrected cell (or condition) means is absolutely necessary.

Is this really true? Why/why not?
The reason I ask this question is because it seems to me that despite this article being published 25 years ago, it seems to me that people generally still generally interpret interactions in the way condemned by Rosnow and Rosenthal.
Is that because a bad habit has persisted? Or were Rosnow and Rosenthal wrong in the first place?
 A: This will not be a full answer, but I'll say a few things.
I am suspicious of any advice that says that such and such is "absolutely necessary." While looking at interaction effects is one way to help understand an interaction, it is not the only possible way, nor may it be the best way, in some contexts.
I also tend to discount advice that does not include the word "graph" or a synonym thereof. We do statistics because we want to understand what is going on in the data. A person who is unduly focused on $F$ and $t$ statistics, $P$ values, and residual effects is looking at trees and probably not appreciating the forest. I think an interaction plot -- or a dozen of them, depending on the complexity of the situation -- is almost always a very good start in understanding an interaction. Even in a case where the interaction $F$ test is not significant, an interaction plot that looks like an interaction is present provides a pretty strong message about the inadequacy of the data to identify it.
So my general answer to the question is to try to really understand what is going on, and use whatever it takes to achieve that understanding.
A: While it looks generally wise to require closer investigation of the data, the authors overlooked important facts that weaken their argumentation. I'll show these fact on the illustrational examples they provide.
1) The first is about the possible interaction of child's sex and health status to "grief" (about the child's death?). This is --as often in psychometrics-- measured on an ordinal, not metric scale. So it is not allowed do calculate differences in grief. (What should if be after all? How far you are bowed down to the floor with grief?) This eradicates a major requisite of the author's argumentation, namely the decomposition of the effects into interaction and main effect by taking differences of the means (also not appropriate in ordinal analysis). In the end, for such an ordinal layout, all you can say is in fact "healthy male > healthy female > sick female > sick male". 
This is in conjunction to a noteworthiness of the term "interaction" in such settings, and another flaw both of the authors and those they criticize. Namely, the only way to prove an $2\times 2$ ordinal interaction to be significant would be a X shaped interaction plot. (Like in Figure 1 in the paper). Why? 
Assume a "<" shaped interaction plot like in the first example (Table 1). As the size of differences on the grief scale, you can choose a monotonic transformation that moves the 3 close to 1 and -1 close to -3. This does not destroy the essential information content in the data. But now you almost have a "=" shaped interaction plot and would conclude (even from ANOVA, which is not appropriate to ordinal data anyway; one should use nonparametric procedures there) that there is no interaction. 
So an ordinal interaction would have been "healthy male >= sick female > healthy female >= sick male". This X-shaped pattern cannot be destroyed by monotonically rescaling the (arbitrary) ordinal scaling of "grief".
2) Concerning the second example, there is a totally different flaw.
This example considers something metric, namly the numbers of hits of baseball players, who were subjected to $2\times 2$ possibly interacting conditions. Now it is OK to calculate differences in hits, and a decomposition into main effect and interaction effect is allowed. But is it unique? 
We can never tell. Consider Table 6:
                       a0               a1
                   b0      b1       b0      b1  
group mean          3       3        5       7 
row effect       -1.5    -1.5      1.5     1.5   
column effect    -0.5     0.5     -0.5     0.5   
grand mean                     4.5
interaction      +0.5    -0.5     -0.5    +0.5  

What makes the authors believe that -1.5 is an unbiased estimator of the row effect of a0 and 1.5 of a1? They chose these values analogously to least squares estimation, but LSE can only estimate the expected value. It cannot tell us how to decompose unknown parameters into even more unknown summands.
And we are interested in these unkown summands! Why is there a colum effect of +/-0.5 between a0b0 and a0b1 if both cell values are exactly the same? It is because of the other cells. That means, due to completely different baseball players, namly those under condition a1, we conclude that if we would treat a player of group a0 with condition b1 instead of b0, he would hit once more per game? Although in group a0 no difference between condition b1 and b0 has been observed? Can this be true? Or is it simply a statistical mirage?
The statistical background of this phenomenon has been discovered by Rao (1962) and is called estimability. It can be shown that in this simple $2\times2$ layout with all four interaction effects, main effects are not estimable, that means they depend on something arbitrary. That causes this mirage. 
The main effect estimators can only become unique if we remove the interactions from the model. So Rosnow and Rosenthal want to compare terms that are simply not present at the same time. 
This error leads also to the erroneous conclusion that significant interactions are always X-shaped.
But they are not completely wrong: If you don't find a significant interaction in ANOVA and want to start considering only the main effects, one should have in mind that a type-II-error could have occured, and that there is in fact an interaction that biases the estimation and tests of the main effects. So an interaction plot with confidence intervals would be a good idea, as it also sheds more light on the effects themselves.
