  A variety of experiments have been carried out to establish the origin of the
chiral charge-density wave transition in 1T-TiSe$_2$, which in turn has led to
contradictory conclusions on the origin of this transition. Some studies
suggest the transition is a phonon-driven structural distortion while other
studies suggest it is an excitonic insulator phase transition that is
accompanied by a lattice distortion. First, we propose these interpretations
can be reconciled if one analyzes the available experimental and theoretical
data within a formal definition of what constitutes an excitonic insulator as
initially proposed by Keldysh and Kopaev. Next, we present pump-probe
measurements of circularly polarized optical transitions and first-principles
calculations where we highlight the importance of accounting for structural
distortions to explain the finite chirality of optical transitions in the CDW
phase. We show that at the elevated electronic temperature that occurs upon
photoexcitation, there is a non-centrosymmetric structure that is
near-degenerate in energy with the centrosymmetric charge density wave
structure, which explains the finite chirality of the optical transitions
observed in the CDW phase of TiSe$_2$.
