  We study weak radiative $|\Delta c|=|\Delta u|=1$ decays of the charmed
anti-triplett ($\Lambda_c$, $\Xi_c^{+}$, $\Xi_c^{0}$) and sextet
($\Sigma_c^{++}$, $\Sigma_c^+$, $\Sigma_c^0$, $\Xi_c^{\prime +}$,
$\Xi_c^{\prime 0}$, $\Omega_c$) baryons in the standard model (SM) and beyond.
We work out $SU(2)$ and $SU(3)_F$-symmetry relations. We propose to study
self-analyzing decay chains such as $\Xi_c^+ \to \Sigma^+ (\to p \pi^0) \gamma$
and $\Xi_c^0 \to \Lambda (\to p \pi^-) \gamma$, which enable new physics
sensitive polarization studies. SM contributions can be controlled by
corresponding analysis of the Cabibbo-favored decays $\Lambda_c^+ \to \Sigma^+
(\to p \pi^0) \gamma$ and $\Xi_c^0 \to \Xi^0 (\to \Lambda \pi^0) \gamma$.
Further tests of the SM are available with initially polarized baryons
including $\Lambda_c \to p \gamma$ together with $\Lambda_c \to \Sigma^+
\gamma$ decays, or $\Omega_c \to \Xi^0 \gamma$ together with $\Omega_c \to
(\Lambda,\Sigma^0) \gamma$. In addition, CP-violating new physics contributions
to dipole operators can enhance CP-asymmetries up to few percent.
