  The two-photon transition $\psi(3686)\to\gamma\gamma J/\psi$ is studied in a
sample of 106 million $\psi(3686)$ decays collected by the BESIII detector. The
branching fraction is measured to be
$(3.1\pm0.6(\unit{stat})^{+0.8}_{-1.0}(\unit{syst})) \times10^{-4}$ using
$J/\psi\to e^+e^-$ and $J/\psi\to\mu^+\mu^-$ decays, and its upper limit is
estimated to be $4.5\times10^{-4}$ at the 90% conference level. This work
represents the first measurement of a two-photon transition among charmonium
states. The orientation of the $\psi(3686)$ decay plane and the $J/\psi$
polarization in this decay are also studied. In addition, the product branching
fractions of sequential $E1$ transitions $\psi(3686)\to\gamma\chi_{cJ},
\chi_{cJ}\to\gamma J/\psi (J=0,1,2)$ are reported.
