  Since their discovery in 2003, the open charm states $D_{s0}^*(2317)$ and
$D_{s1}(2460)$ provide a challenge to the conventional quark model. In recent
years, theoretical evidence has been accumulated for both states in favor of a
predominantly $DK$ and $D^*K$ molecular nature, respectively. However, a direct
experimental proof of this hypothesis still needs to be found. Since radiative
decays are generally believed to be sensitive to the inner structure of the
decaying particles, we study in this work the radiative and strong decays of
both the $D_{s0}^*(2317)$ and $D_{s1}(2460)$, as well as of their counterparts
in the bottom sector. While the strong decays are indeed strongly enhanced for
molecular states, the radiative decays are of similar order of magnitude in
different pictures. Thus, the experimental observable that allows one to
conclusively quantify the molecular components of the $D_{s0}^*(2317)$ and
$D_{s1}(2460)$ is the hadronic width, and not the radiative one, in
contradistinction to common belief. We also find that radiative decays of the
sibling states in the bottom sector are significantly more frequent than the
hadronic ones. Based on this, we identify their most promising discovery
channels.
