  The mass spectra, mixing angle and decay constants of the $J^P=1^+$
heavy-light mesons are systematically studied within the framework of the
Bethe-Salpeter equation (BSE). The full $1^+$ Salpeter wave function is given
for the first time. The mixing between the $1^{+-}$ and $1^{++}$ in the $1^+$
heavy-light systems are automatically determined by the dynamics in the
equation without any man-made mixing. The results indicate that in a rigorous
study there exists the phenomenon of mixing angle inversion or mass inversion
within $1^{+}$ heavy-light doublet, which is sensitive to the $s$-quark mass
for the charmed mesons and $u$- or $d$-quark masses for the bottomed mesons.
This inversion phenomenon can answer the question of why we have confused
mixing angles in the literature and partly explain the lower mass of
$D_{s1}(2460)$ compared to that of $D_{s1}(2536)$. The decay constants are also
presented and can behave as a good quantity to distinguish the $1^+$ doublet in
heavy-light mesons. This study indicates that the light-quark mass may play an
important role in deciding the mass order, mixing angle, and decay constant
relation between the $\ket{j_l=\frac{3}{2}}$ and $\ket{j_l=\frac{1}{2}}$
heavy-light mesons.
