We have 2 continuous predictors in a regression model, and both have significant quadratic terms. Our main hypothesis we'd like to test is (generally) whether there is an interaction between these variables. When modelling, we see a significant interaction between the linear terms. But interactions terms involving the quadratic terms are not signficant.
So are there any problems with fitting the model with the quadratic main effects and only first order interaction terms (e.g. $y = intercept + x^2 + x + xz + z + z^2$)? I don't know of any theoretical problems with this, but looking at the predicted outcome, this seems to force a strange relationship -- specifically, looking at curves of $y$ versus $x$, for all values of $z$ the curves intersect at a specific point.
In short, the question is, if you are set on including a quadratic covariate, are you then only interested in testing interactions of the highest ordered terms (and if only linear interactions are significant then this means there is no interaction.