  Fluctuations arise universally in Nature as a reflection of the discrete
microscopic world at the macroscopic level. Despite their apparent noisy
origin, fluctuations encode fundamental aspects of the physics of the system at
hand, crucial to understand irreversibility and nonequilibrium behavior. In
order to sustain a given fluctuation, a system traverses a precise optimal path
in phase space. Here we show that by demanding invariance of optimal paths
under symmetry transformations, new and general fluctuation relations valid
arbitrarily far from equilibrium are unveiled. This opens an unexplored route
toward a deeper understanding of nonequilibrium physics by bringing symmetry
principles to the realm of fluctuations. We illustrate this concept studying
symmetries of the current distribution out of equilibrium. In particular we
derive an isometric fluctuation relation which links in a strikingly simple
manner the probabilities of any pair of isometric current fluctuations. This
relation, which results from the time-reversibility of the dynamics, includes
as a particular instance the Gallavotti-Cohen fluctuation theorem in this
context but adds a completely new perspective on the high level of symmetry
imposed by time-reversibility on the statistics of nonequilibrium fluctuations.
The new symmetry implies remarkable hierarchies of equations for the current
cumulants and the nonlinear response coefficients, going far beyond Onsager's
reciprocity relations and Green-Kubo formulae. We confirm the validity of the
new symmetry relation in extensive numerical simulations, and suggest that the
idea of symmetry in fluctuations as invariance of optimal paths has
far-reaching consequences in diverse fields.
