The reason for centering a continuous covariate is that it can improve interpretability.
For example, say you had one categorical predictor with 4 categories and one continuous covariate, plus an interaction between them.
First, you’ll notice that if you center your covariate at the mean, there is no difference in the ANOVA table (Tests of Between Subjects Effects). There may be tiny differences due to rounding, but the results should generally not change.
The categorical variable would be interpreted by looking at the estimated marginal means (in SPSS; in SAS, they are called least squares means). These means are interpreted as the means of the DV at the mean value of the covariate. If you centered your covariate at its mean, there should be no difference whatsoever in the estimated marginal means.
The covariate and interaction would be interpreted by looking at the Parameter Estimates table.
The B values are the regression coefficients (slopes). With an interaction in the model, the B value for the covariate is the slope when the categorical variable =0. The Parameter Estimates table automatically dummy codes your categorical variables which means it makes the category that comes last alphabetically = 0 (If you numbered them 1,2,3,4, then 4 comes last alphabetically–you can change this default when you run the GLM). So the B value for the covariate is the slope of the covariate only for group 4.
The B values listed for the interactions with the other groups are the differences in the slopes between each of those groups and group 4. If those B values are significant, you know that their slopes are significantly different than group 4’s slope.
Again, these don’t change whether the covariate is centered or not.
What does change are the intercepts. And you now have 4 intercepts (one line for each category). The B labeled Intercept in the output is the intercept just for the reference category (group 4). The Bs for the other three groups are the differences in intercepts between group 4 and each of those groups.
Remember the intercept is the mean of the dependent variable when the covariate = 0. When you center the covariate, you are changing the 0 point. So the intercepts are no longer the average value when Covariate=0 on its original scale but the average value when the Covariate is at its mean.
This is especially helpful when the Covariate never has values even close to 0. For example, if your Covariate was Age, and your Ages ranged from 20-60, the mean value of the DV at birth doesn’t make much sense.