  The interaction of the solar wind with the Earth's magnetic field produces
geomagnetic activity, which is critically dependent on the orientation of the
interplanetary magnetic field (IMF). Most solar wind coupling functions
quantify this dependence on the IMF orientation with the so-called IMF clock
angle in a way, which is symmetric with respect to the sign of the $B_y$
component. However, recent studies have suggested that the sign of $B_y$ is an
additional, independent driver of high-latitude geomagnetic activity, leading
to higher (weaker) geomagnetic activity in Northern Hemisphere (NH) winter for
$B_y\! >\! 0$ ($B_y\! <\! 0$). In this paper we quantify the size of this
explicit $B_y$-effect with respect to the solar wind coupling function, both
for Northern and Southern high-latitude geomagnetic activity. We show that
high-latitude geomagnetic activity is significantly (by about 40-50\%)
suppressed for $B_y\!<\!0$ in NH winter and for $B_y\!>\!0$ in SH winter. When
averaged over all months, high-latitude geomagnetic activity in NH is about
12\% weaker for $B_y\!<\!0$ than for $B_y\!>\!0$. The $B_y$-effect affects the
westward electrojet strongly but hardly at all the eastward electrojet. We also
show that the suppression of the westward electrojet in NH during $B_y\!<\!0$
maximizes when the Earth's dipole axis points towards the night sector, i.e.,
when the auroral region is maximally in darkness.
