We derive various exact results for Markovian systems that spontaneously relax to a non-equilibrium steady-state by using joint probability distributions symmetries of different entropy production decompositions. The analytical approach is applied to diverse problems such as the description of the fluctuations induced by experimental errors, for unveiling symmetries of correlation functions appearing in fluctuation-dissipation relations recently generalised to non-equilibrium steady-states, and also for mapping averages between different trajectory-based dynamical ensembles. Many known fluctuation theorems arise as special instances of our approach, for particular two-fold decompositions of the total entropy production. As a complement, we also briefly review and synthesise the variety of fluctuation theorems applying to stochastic dynamics of both, continuous systems described by a Langevin dynamics and discrete systems obeying a Markov dynamics, emphasising how these results emerge from distinct symmetries of the dynamical entropy of the trajectory followed by the system For Langevin dynamics, we embed the "dual dynamics" with a physical meaning, and for Markov systems we show how the fluctuation theorems translate into symmetries of modified evolution operators. ; Comment: 39 pages, 1 figure. Minor revision, as suggested by referees. A couple of references and equations added. Acknowledgements slightly modified