The antiparallel reconnection hypothesis states that reconnection occurs at or near the locus along which magnetosheath and magnetospheric fields are antiparallel and the reconnecting component of the magnetic field is maximal. Recent observations have pointed toward significant reconnection rates in locales with much less than antiparallel field shear. We explore the alternative component reconnection hypothesis that reconnection occurs along a locus determined by integrating the local X line (XL) direction away from that region with the largest reconnecting field magnitude. We develop a description of the magnetospheric and magnetosheath magnetic fields at the magnetopause, compute the angle between the sheath and boundary layer fields, and the reconnecting component, everywhere on the magnetopause. We then integrate the XL across the magnetopause from a starting point or points where the reconnecting component is maximal. The resultant XL lies across the equatorial subsolar magnetopause for southerly Bz, as expected. For typical interplanetary magnetic field (IMF) conditions dominated by By, the XL tilts poleward and loops around the cusps into the high-latitude regions, consistent with throat-like cusp region flows. For northerly Bz, the XL bifurcates into high-latitude and low-latitude segments with subsolar reconnection disappearing only for mainly northward Bz. We argue that the component reconnection XL represents the mean location of quasi-steady reconnection on the magnetopause, subject to of course to nonuniformities and variations of the interplanetary magnetic field and plasma. We conclude that reconnection should have appreciable rates across the subsolar magnetopause for most IMF clock angles, with higher rates at high latitudes for northward Bz.