  Modern observatories of ultra-high energy cosmic rays (UHECR) have collected
over 10^4 events with energies above 10 EeV, whose arrival directions appear to
be nearly isotropically distributed. On the other hand, the distribution of
matter in the nearby Universe -- and, therefore, presumably also that of UHECR
sources -- is not homogeneous. This is expected to leave an imprint on the
angular distribution of UHECR arrival directions, though deflections by cosmic
magnetic fields can confound the picture. In this work, we investigate
quantitatively this apparent inconsistency. To this end we study observables
sensitive to UHECR source inhomogeneities but robust to uncertainties on
magnetic fields and the UHECR mass composition. We show, in a rather
model-independent way, that if the source distribution tracks the overall
matter distribution, the arrival directions at energies above 30 EeV should
exhibit a sizeable dipole and quadrupole anisotropy, detectable by UHECR
observatories in the very near future. Were it not the case, one would have to
seriously reconsider the present understanding of cosmic magnetic fields and/or
the UHECR composition. Also, we show that the lack of a strong quadrupole
moment above 10 EeV in the current data already disfavours a pure proton
composition, and that in the very near future measurements of the dipole and
quadrupole moment above 60 EeV will be able to provide evidence about the UHECR
mass composition at those energies.
