There lies information in a discrepancy of the (unconditional) mean and median. For example, if the median is larger than the mean, the distribution must be left-skewed.
Does this kind of inference translate to conditional means and medians as estimated by ordinary least squares and quantile regression?
For example, if the quantile regression coefficient is larger than the OLS coefficient, can we say that the conditional distribution of Y given X is left-skewed?
Is any such or related interpretation valid in the presence of multiple independent variables?