Charge density waves (CDWs) with multi-component order parameters can break unexpected symmetries through the interplay of nearly degenerate instabilities. In 1T-TiSe₂, whether the CDW spontaneously breaks mirror and inversion symmetries to form a chiral state has remained controversial, with previous experiments reporting conflicting results. We resolve this controversy by identifying the bulk broken symmetry as ferroaxial: vertical mirrors are broken while inversion is preserved. Symmetry-resolved elastoresistivity reveals off-diagonal coefficients forbidden in both the high-temperature phase and the widely accepted triple-Q CDW state, providing direct evidence that mirror symmetries are broken below T_CDW ~ 200 K while inversion is retained. The diagonal coefficients show a divergent nematic susceptibility only at a much lower temperature, T_nem ~ 165 K, establishing that nematic order is not responsible for the mirror symmetry breaking near T_CDW. Elastocaloric measurements resolve two successive transitions: a primary CDW onset and a secondary transition approximately 7 K below, consistent with the condensation of a secondary CDW mode required to produce ferroaxial order. Together, these data establish a hierarchy of symmetry-breaking transitions in which the primary CDW is followed by a ferroaxial instability, which in turn acts as the parent state for a secondary nematic instability. The proposed "chiral" CDW in 1T-TiSe₂ is thus a centrosymmetric ferroaxial state, reconciling previous surface-sensitive observations with bulk symmetry constraints.