Why are my regression results different depending on reference level of dichotomous variables? I'm running some regression models, but I'm finding that my pattern of results change depending on the reference groups. My predictors are Condition(A vs. B), Status (low vs. high), and Age (continuous, mean-centered).
In R, my first model is:  score ~ condition x status x age
Results when reference groups are condition = A, status = low:

Results when reference groups are condition = A, status = high:

Results when reference groups are condition = B, status = low:

Results when reference groups are condition = B, status = high:

Plot of the raw data:

I understand that changing the reference groups will change the coefficients, but I assumed the t-values and p-values would remain the same since I have dichotomous variables. There isn't a strong theoretical basis for using one reference group over the other. Looking at a plot of the raw data, it looks like there should at least be main effects of condition, status, and age, as well as an interaction of condition*status. Why am I getting different results and how might I address this issue?
I'm also running these same models on different DVs and sometimes the results are the same regardless of the reference groups and sometimes they differ.  I have this same issue in other mixed-effects models where I just have an addition dichotomous within-group predictor so I have the participant ID included as a random effect.
 A: 
I understand that changing the reference groups will change the coefficients, but I assumed the t-values and p-values would remain the same since I have dichotomous variables.

The coefficient t-tests are for determining whether the estimated value of a coefficient is significantly different from 0. That's not a test of the overall significance of a predictor.
When a predictor is involved in interactions, its individual coefficient values and those for its lower-level interactions are estimated for a situation when all its interacting categorical predictors are at their own reference levels.* So changing the reference level of a categorical predictor can change the value of coefficients involving other predictors with which it interacts, and thus the "significance" of their differences from 0.
Despite the apparent differences in coefficient "significance," any predictions made from those models will be identical. The models are fundamentally the same. Only the ways that initial summaries are shown are different. If you want to evaluate the overall significance of any predictor, you can do likelihood-ratio tests of nested models with and without it, or perform Wald tests that evaluate together all coefficients involving it.

*This explanation is for the default R treatment/dummy coding of predictors.
