ISEE 1 and 2 magnetic field data obtained during dayside magnetopause crossings have been analyzed using a minimum variance technique in order to determine the characteristics of the rotation of the magnetic field across the magnetopause. Only a small fraction of the magnetopause crossings examined present a suficiently coherent structure that they can be compared with theoretical predictions, and only a few of the hodograms of the magnetic field at the crossings show the ideal rotation expected for a rotational discontinuity structure. Observed senses of rotation are not consistent with the predictions of the electron whistler polarization theory. Rotations greater than 180¿ are very seldom observed; they usually occur across multiple crossings, either as 360¿ rotation about the minimum variance direction or, occasionally in more complicated patterns associated with 'S'-shaped hodograms. These observations suggest that the sense of the magnetic field rotation through the magnetopause is controlled by the relative orientation of the magnetosheath and magnetospheric field and that the angular rotation is minimized when the magnetic field changes from one orientation to the other.