  The first law of thermodynamics imposes not just a constraint on the
energy-content of systems in extreme quantum regimes, but also
symmetry-constraints related to the thermodynamic processing of quantum
coherence. We show that this thermodynamic symmetry decomposes any quantum
state into mode operators that quantify the coherence present in the state. We
then establish general upper and lower bounds for the evolution of quantum
coherence under arbitrary thermal operations, valid for any temperature. We
identify primitive coherence manipulations and show that the transfer of
coherence between energy levels manifests irreversibility not captured by free
energy. Moreover, the recently developed thermo-majorization relations on
block-diagonal quantum states are observed to be special cases of this symmetry
analysis.
