The theory of resonant modes is extended to finite length systems containing pinch points of complex axial wavenumber k0 and frequency ω0 with arbitrary ωkk0=∂2ω/∂k2. The quantity ωkk0 is shown to be an important indicator of how streamwise boundary conditions modify the local absolute mode at (k0,ω0). In particular, when Im(ωkk0)>0, the pinch point is twisted, and resonant modes owing to streamwise boundary conditions may then have growth rates greater than that of the unbounded absolute mode. In this case, global instability may occur while the flow is only convectively unstable. The premixing zone between the nozzle and a lifted flame on a variable-density jet is an example of a streamwise-confined system containing a twisted pinch point. For this system, linear stability analysis is employed to locate resonant modes along a solution curve in the complex k and ω planes. The orientation of the solution curve predicts destabilization owing to streamwise confinement as well as increasing global frequency with decreasing lift-off height as observed in previous direct numerical simulations. The theory also suggests that low-frequency fluctuations observed in the simulations may be explained by beating between two resonant modes of slightly differing frequencies.