This work attempts to unveil the similarities and differences between Jahn–Teller (JT) and non-JT systems involving CuF₆⁴⁻ units. For achieving this goal, we firstly explore Na₂CuF₄ and NaF:Cu²⁺ systems through first principles calculations and pay particular attention to the links between JT and non-JT systems looking at the electronic density of the hole. The results on Na₂CuF₄ in the monoclinic P2₁/c space group and also in the parent Pbam structure reveal that the local geometry can be understood as an initial tetragonally compressed CuF₆⁴⁻ unit, followed by an additional orthorhombic instability that excludes the JT effect as the origin. Although the present results on NaF:Cu²⁺ underpin an elongated equilibrium geometry such as that measured for Cu²⁺ ions in the cubic perovskite KZnF₃, the force constant for NaF:Cu²⁺ is half that for KZnF₃:Cu²⁺. This crucial fact is direct proof of the elastic decoupling of CuF₆⁴⁻ from the NaF lattice leading to a JT energy, E_JT, which is twice that found for KZnF₃:Cu²⁺. However, both systems have practically the same linear electron–vibration coupling constant, V_1e, a relevant fact whose origin is discussed. The final aim of this work concerns the influence of tetragonal and orthorhombic distortions as well as the internal electric field on the A_1g–B_1g energy gap, Δ, of a variety of systems with CuF₆⁴⁻ complexes. Interestingly, it is shown that compounds with orthorhombic instability and an internal electric field can have a Δ value comparable to the JT system NaF:Cu²⁺. Accordingly, explanations for optical spectra of transition metal compounds based on simple parameterized models can be meaningless. The present study shows that properties displayed by d⁹ compounds in low symmetry lattices can hardly stem from a static JT effect.