  Using new experimental data on photoproduction and $\Lambda_b^0$ decays, we
derive constraints on the properties of the LHCb $P_c$ states. We conclude that
$P_c(4312)$, $P_c(4380)$ and $P_c(4440)$ can be described as $\Sigma_c\bar D$,
$\Sigma_c^*\bar D$ and $\Sigma_c\bar D^*$ molecules, but that $P_c(4457)$ does
not fit into the same picture. Based on the apparent absence of additional
partner states, and the striking disparity between $\Lambda_c^+\bar D^0$ and
$\Lambda_c^+\bar D^{*0}$ decays, we conclude that $P_c(4440)$ has $3/2^-$
quantum numbers. Using heavy-quark symmetry we predict large experimental
signals for $P_c$ states in $\eta_c p K^-$, $\Lambda_c^+\bar D^{*0}K^-$, and
$\Sigma_c^{(*)}\bar D K^-$. We also argue that current experimental data on
photoproduction is almost at the level of sensitivity required to observe $P_c$
states.
