  Inclusive J/$\psi$ production has been studied with the ALICE detector in
p-Pb collisions at the nucleon-nucleon center of mass energy $\sqrt{s_{\rm
NN}}$ = 5.02 TeV at the CERN LHC. The measurement is performed in the center of
mass rapidity domains $2.03<y_{\rm cms}<3.53$ and $-4.46<y_{\rm cms}<-2.96$,
down to zero transverse momentum, studying the $\mu^+\mu^-$ decay mode. In this
paper, the J/$\psi$ production cross section and the nuclear modification
factor $R_{\rm pPb}$ for the rapidities under study are presented. While at
forward rapidity, corresponding to the proton direction, a suppression of the
J/$\psi$ yield with respect to binary-scaled pp collisions is observed, in the
backward region no suppression is present. The ratio of the forward and
backward yields is also measured differentially in rapidity and transverse
momentum. Theoretical predictions based on nuclear shadowing, as well as on
models including, in addition, a contribution from partonic energy loss, are in
fair agreement with the experimental results.
