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Black holes in four-dimensional asymptotically flat general relativity possess vanishing static tidal Love numbers (TLNs), a remarkable property whose underlying origin has remained elusive. Recent works identified a hidden algebraic structure in the perturbation equations, known as Ladder symmetry, which guarantees the absence of static scalar TLNs for a subclass of spacetimes within the Konoplya–Rezzolla–Zhidenko (KRZ) parametrization. In this work, we investigate whether Ladder symmetry is not only sufficient, but also necessary for the vanishing of static TLNs. We introduce parametric deformations away from Ladder-symmetric backgrounds and analyze the resulting linear tidal response using the parametrized formalism for TLNs. We show that any deviation from a Ladder-symmetric spacetime generically produces non-zero static scalar TLNs. Our results establish Ladder symmetry as both a necessary and sufficient condition for the vanishing of static TLNs in static, spherically symmetric black holes and in rotating black holes within the KRZ class