The theory of the Jahn-Teller effect is treated for the case of trigonal symmetry and strong vibronic interaction. The vibronic Hamiltonian is established assuming that the vibrational modes are any E modes whatsoever but respecting the trigonal symmetry. The trigonal potential and the vibronic interaction are diagonalized at the same time, in the basis of the cubic T₁ or T₂ electronic states. It is shown that the lowest energy vibronic levels are a doublet and a singlet. The sign of the splitting between these two levels can be different from the sign of the electronic trigonal splitting. Therefore, in the general case, the apparent quenching of the trigonal potential is different from the quenching of the spin-orbit coupling. This theory for trigonal symmetry is also valid in the cubic limit. In particular it allows us to determine the quenching of the trigonal splitting. This quenching factor is equal to 8/9 for a strong Jahn-Teller effect with T_2g modes. This model is compared to earlier models applied to the (3d)ⁿ ions in α-Al₂0₃.