The coding you are doing is called effects coding (centering the Dummy coding). The interpretation of the coefficients is different when you use effects coding, which will cause the correlation with the coefficients to change.
This FAQ from UCLA summarizes the problem better than I can
Why use effect coding?
Here’s a good question, why use effect coding instead of dummy coding?
If you have several categorical variables in a model it often doesn’t
make much difference whether you use effect coding or dummy coding.
However, if you have an interaction of two categorical variables then
effect coding may provide some benefits. The primary benefit is that
you get reasonable estimates of both the main effects and interaction
using effect coding. With dummy coding the estimate of the interaction
is fine but main effects are not “true” main effects but rather what
are called simple effects, i.e., the effect of one variable at one
level of the other variable. This is why most analysis of variance
programs use some type of effect coding when estimating the various
effects in an ANOVA model.
In terms of how it can effect correlation, you can look at the interaction for dummy coding vs effects coding.
In dummy coding, the coefficients for Positive Positive condition is for 1*1=1. This will be run in contrast to the coefficients for Negative Negative condition 0*0=0 which is the same as Negative Positive(1*0=0) and Positive Negative (0*1=0).
In Effects coding, the coefficients for Positive Positive 0.5*0.5 = 0.25 and Negative Negative = -0.5*-0.5 = 0.25 will be computed together against the Negative Postive and Positive Negative (-0.5*0.5=0.5*-0.5 = -0.25) conditions.
Since the main effect and interaction coefficients in effects coding are estimating different things than dummy coding (and they are interpreted differently), there is no reason why the correlation would remain the same.
Whether effects coding should be done or not for mixed-effect regression depends on the research intent, which is the same as for regular regression. Effects and dummy coding carry the same information, it's a matter of interpretation that determines which to use.