Centering variables involved in an interaction changed results, how is this correctly interpreted? In my model, some potentially confounding continuous variables had to be taken in account as well as an interaction of this variable with the main factor of interest, which is categorical. Initially, I had been doing some modelling in R but I switched to JMP for some extra plots. There the p-values of my main factor of interest changed dramatically. After some digging, I found out that JMP centers variables when they are involved in an interaction and that this option could be disabled. As expected, the p-values agreed with the R output as soon as I disabled it. I found an explanation as to why the p-values change when an interacting variable is centered here: p-values change after mean centering with interaction terms. How to test for significance?
However, while this explains how it could happen, I'm still in the dark as to how you could then decide which is more relevant? Somewhere in the link it is noted that the correct test for significance should involve all coefficients of the interacting effects at once but I'm completely unaware of how to do that?

The coefficient reported for an individual predictor by most statistical software needs careful interpretation when the predictor is involved in interactions. The coefficient is for its association with outcome when all of its interacting predictors have values of 0 (continuous interactors) or are at reference levels (categorical interactors). The p-values evaluate whether the coefficient value under those conditions differs "significantly" from 0. When there's an interaction, the value of an individual coefficient estimate and its apparent "significance" of a difference from 0 (in that highly restricted interpretation) thus changes as you center the interacting predictors.
Nevertheless, as you found, the model is fundamentally the same whether or not you center the predictors. All model predictions would be the same either way. I that context, I'd say that no individual coefficient is "relevant" on its own (except for the coefficient for the very highest level of interactions, which isn't affected by centering).
Testing whether a predictor including its interactions has a "significant" association with outcome requires a combined test of some type. These tests typically have a name like "anova" although they apply to situations beyond the classic ANOVA situation with categorical predictors and continuous outcomes.
For a single predictor, you can fit two models: one with it (and its interactions), one without, and use anova() to perform a likelihood-ratio test between the two models. Alternatively, without fitting an additional model, you can perform what's called a Wald test on the entire set of coefficients involving the predictor. That takes into account those coefficients and their standard errors and the covariances among the coefficient estimates. That's the default for the Anova() function (note the capital "A") in the R car package and for anova() when applied to models built with the R rms package. Be careful in using the basic R anova() function on a single model, however, as its default can be misleading when the study isn't completely balanced.
