  The spectrum of the charmed meson-(anti)meson system is a fundamental tool
for disentangling the nature of a few exotic hadrons, including the recently
discovered $T_{cc}^+(3875)$ tetraquark, the $X(3960)$, or the $X(3872)$, the
nature of which is still not clear after almost two decades of its discovery.
Here we consider that the charmed meson-(anti)meson short-range interaction is
described by the exchange of light-mesons ($\sigma$, $\rho$, $\omega$). The
effects of light-meson exchanges are recast into a simple contact-range theory
by means of a saturation procedure, resulting in a compact description of the
two-hadron interaction. From this, if the $T_{cc}^+$ were to be an isoscalar
$D^* D$ molecule, then there should exist an isoscalar $J=1$ $D^* D^*$ partner,
as constrained by heavy-quark spin symmetry. Yet, within our model, the most
attractive two charmed meson configurations are the isovector $J=0$ $D^* D^*$
molecule and its sextet $D_s^* D^*$ and $D_s^* D_s^*$ flavor partners. Finally,
we find a tension between the molecular descriptions of the $T_{cc}^+$ and that
of the $X(3872)$ and $X(3960)$, where most parameter choices suggest that if
the $T_{cc}^+$ is purely molecular then the $X(3872)$ overbinds (or conversely,
if the $X(3872)$ is a molecule the $T_{cc}^+$ does not bind). This might be
consequential for determining the nature of these states.
