  Ta$_2$NiSe$_5$ is one of the most promising materials for hosting an
excitonic insulator ground state. While a number of experimental observations
have been interpreted in this way, the precise nature of the symmetry breaking
occurring in Ta$_2$NiSe$_5$, the electronic order parameter, and a realistic
microscopic description of the transition mechanism are, however, missing. By a
symmetry analysis based on first-principles calculations, we uncover the
\emph{discrete} lattice symmetries which are broken at the transition. We
identify a purely electronic order parameter of excitonic nature that breaks
these discrete crystal symmetries and contributes to the experimentally
observed lattice distortion from an orthorombic to a monoclinic phase. Our
results provide a theoretical framework to understand and analyze the excitonic
transition in Ta$_2$NiSe$_5$ and settle the fundamental questions about
symmetry breaking governing the spontaneous formation of excitonic insulating
phases in solid-state materials.
