Early satellite observations of the dayside magnetopause have suggested that the magnetic field typically rotates clockwise above the solar-magnetospheric equatorial plane and counterclockwise below it, in agreement with the predictions of first-order orbit theory for magnetopause crossings of the rotational discontinuity type. The present paper treats the tangential discontinuity (TD) case. The influence of magnetosheath magnetic field and plasma flow on the magnetopause equilibrium structure is analyzed by means of a Vlasov model. The nature of the current layer plays a major role; the analysis is carried out for ion-dominated, electron-dominated, and mixed layers. Necessary and sufficient conditions for the existence of an equilibrium magnetopause are derived. It is found that (1) the magnetopause is best modeled as a transition layer of mixed type; (2) for high magnetic shear the magnetic field at the dayside magnetopause preferentially rotates clockwise above the equatorial plane and counterclockwise below it, while it rotates counterclockwise above and clockwise below the equatorial plane at the tail flanks; this effect becomes more manifest as the magnetosheath flow is faster and as the difference in proton and electron transition lengths is more pronounced; (3) for low magnetic shear, TD equilibrium is expected to be lost more easily at the dawnside than at the duskside; and (4) the model provides a magnetopause thickness estimate; in particular, the low magnetic shear dawn magnetopause is predicted to be thinner than the dusk magnetopause.