With balanced data and sum-to-zero contrasts, the intercept is indeed the general mean of the response; the coefficient of a1 is the mean of the response in category a1 minus the general mean; the coefficient of a1:b1 is the mean of the response in cell a1, b1 minus the general mean and the coefficients of a1 and b1; etc. For unbalanced data (and balanced data) the intercept is the mean of the cell means; the coefficient of a1 is the mean of cell means at level a1 minus the intercept; etc. Whether all this is of interest is another question, since a simple graph of cell means tells a more digestible story about the data.(c) John Fox, про интерпретацию матрицы регрессионной модели