The vibronic structure of the photoelectron spectra of the X (2)Pi state of XCN(+) (X=F, Cl, and Br) has been calculated, assuming that the X (2)Pi state can be considered as an isolated electronic state. The Renner-Teller coupling of the two components of the (2)Pi state via the degenerate bending mode as well as spin-orbit coupling effects are taken into account. The two stretching modes are treated within the so-called linear vibronic-coupling model. The vibronic and spin-orbit parameters have been determined by accurate ab initio electronic-structure calculations. While spin-orbit effects are small in FCN(+), the large spin-orbit splitting of the X (2)Pi state of the BrCN(+) leads to a complete quenching of the Renner-Teller effect. The X (2)Pi state of the ClCN(+) is shown to be of particular interest: here the resonance condition for linear-relativistic Renner-Teller coupling is approximately fulfilled. This coupling mechanism leads to a significant intensity transfer to vibronic levels with odd quanta of the bending mode. The calculated spectrum indicates that this novel relativistic vibronic-coupling effect should be observable in high-resolution (electron energy resolution of the order of a few meV) photoelectron spectra of ClCN.