  We study the symmetry of large deviation functions associated with
time-integrated currents in Markov pure jump processes. One current known to
have this symmetry is the fluctuating entropy production and this is the
content of the fluctuation theorem. Here we obtain a necessary condition in
order to have a current different from entropy with a symmetric large deviation
function. This condition is related to degeneracies in the set of increments
associated with fundamental cycles from Schnakenberg network theory. Moreover
we consider 4-states systems where we explicitly show that non-entropic
time-integrated currents can be symmetric. We also show that these new
symmetries, as is the case of the fluctuation theorem, are related to
time-reversal. However, this becomes apparent only when stochastic trajectories
are appropriately grouped together.
