Interpreting corr. betw fixed effect & explanatory variable

[Wecon]

Hi,

I'm doing a fixed effects (cross-sectional fixed effects) estimation and since fixed effects estimations discard the information in cross-sectional differences by demeaning the data, I checked the correlations between the fixed effects and the average of my explanatory variables. I found that the estimated fixed effects are significantly correlated with one of the explanatory variables, let's call it x1. More precisely, I calculated, for each cross-section, the average value for x1 over time. It is this over-time-average of x1 that is correlated with the fixed effect. So I wondered: doesn't this correlation provide extra information? Intuitively, I would conclude that this correlation can be interpreted as: cross-sections with an on average higher value for x1, tend to have, on average, a higher value for y (the dependent variable). Nevertheless I cannot find confirmation for this intuition. Is this intuition correct? If not, are there any references or explanations that clarify why it is incorrect?