It is shown that in linear molecules the pseudo-Jahn-Teller (PJT) interaction of a Σ or Π term with a Δ term induces a bending instability that is angular dependent, reducing the symmetry of the adiabatic potential energy surface from expected D∞h to D4h and C∞v to C2v or C4v. This spontaneously broken cylindrical symmetry (BCS) emerges from the solution of the vibronic coupling equations of the PJT effect (PJTE) problems (Σ+Δ)⊗w, (Π+Δ)⊗w, (Π+Σ+Δ)⊗w, and (Δ+Δ)⊗w, where w includes linear, quadratic, and fourth order vibronic coupling terms, and it is confirmed by ab initio calculations for a series of triatomic molecules with ground or excited Δ terms. The BCS is due to the angular symmetry of the electronic wave functions of the Δ term, ∼cos 2φ, and ∼sin 2φ, split by the fourth order vibronic coupling, which in overlap with the other symmetry wave functions of the Σ or Π term provides for the periodical symmetry of the added covalency that facilitates the bending. The mechanism of this PJT-induced BCS effect is discussed in detail; the numerical values of the vibronic coupling parameters for the molecules under consideration were estimated by means of combining separate ab initio calculations of some of them with a procedure fitting the analytical expressions to ab initio calculated energy profiles. It is also shown that the bending of linear molecules in Δ states, similar to Π states, is exclusively a PJT (not Renner-Teller) effect. The BCS revealed in this paper illustrates again the predicting power of the PJTE.