  We extend previous work to describe a class of fluctuation relations (FRs)
that emerge as a consequence of symmetries at the level of stochastic
trajectories in Markov chains. We prove that given such a symmetry, and for a
suitable dynamical observable, it is always possible to obtain a FR under a
biased dynamics corresponding to the so-called generalized Doob transform. The
general transformations of the dynamics that we consider go beyond
time-reversal or spatial isometries, and an implication is the existence of FRs
for observables irrespective of their behaviour under time-reversal, for
example for time-symmetric observables rather than currents. We further show
how to deduce in the long-time limit these FRs from the symmetry properties of
the generator of the dynamics. We illustrate our results with four examples
that highlight the novel features of our work.
