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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?

miura
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    "For example, if the median is larger than the mean, the distribution must be left-skewed." -- that depends on what your definition of skewed is.

    Further the relative weight that is given to observations is different, which makes determining the relative behaviour in all circumstances difficult (e.g. if there are outliers in just the right places, and the actual relationship is curved, and the error distribution has just the right shape, might the expected relative positions be reversed? ) It may be difficult to answer in general.

     – Glen_b Mar 10 '14 at 10:58

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I don't think you can infer much from a difference; the models are fundamentally different. OLS regression makes assumptions that quantile regression does not and, as @glen_b points out, even the direction of skewness isn't completely straightforward.

Further, if there is more than one independent variable, then the size of one coefficient will be affected by the other coefficients; even with only one variable, the intercept may be different.

One thing you could do is plot the predicted median from quantile regression vs. the predicted mean from OLS. You could also plot the differences between the predicted and actual values for each model. It's hard to say what insight this might yield but they are easy plots to draw, so why not?

Peter Flom
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    That is very instructive, and you can also use a model such as an ordinal response model on continuous $Y$ and plot the predicted mean vs. the predicted median from that model, is ordinal models do not force a linear relationship between the two as OLS on $Y$ or $\log(Y)$ (on the anti-log scale) do. Ordinal models, which are semi-parametric (the most popular being the proportional odds and proportional hazards models) are more flexible. – Frank Harrell Mar 10 '14 at 12:24