  In many theories of physics beyond the Standard Model, from extra dimensions
to Hidden Valleys and models of dark matter, Z' bosons mediate between Standard
Model particles and hidden sector states. We study the feasibility of observing
such hidden states through an invisibly decaying Z' at the LHC. We focus on the
process pp -> \gamma Z' -> \gamma X X*, where X is any neutral, (quasi-) stable
particle, whether a Standard Model (SM) neutrino or a new state. This
complements a previous study using pp -> Z Z' -> l+ l- X X*. Only the Z' mass
and two effective charges are needed to describe this process. If the Z' decays
invisibly only to Standard Model neutrinos, then these charges are predicted by
observation of the Z' through the Drell-Yan process, allowing discrimination
between Z' decays to SM neutrinos and invisible decays to new states. We
carefully discuss all backgrounds and systematic errors that affect this
search. We find that hidden sector decays of a 1 TeV Z' can be observed at 5
sigma significance with 50 fb^{-1} at the LHC. Observation of a 1.5 TeV state
requires super-LHC statistics of 1 ab^{-1}. Control of the systematic errors,
in particular the parton distribution function uncertainty of the dominant Z
\gamma background, is crucial to maximize the LHC search
