The Regge spectrum generated by a four-dimensional δ-shell interaction V(r)=λδ(r-a), where r is the four-dimensional radius, is investigated by means of exact solutions of the Wick-rotated Bethe-Salpeter equation. In this model only the leading trajectory can generate resonances. It is infinitely rising with ImαReα<1. Odd daughter trajectories either develop negative imaginary parts or do not rise. Even daughter trajectories turn over above the elastic threshold. This spectrum is contrasted with that obtained from a δ-shell interaction in potential theory. The potential-theory model is characterized by an infinite set of parallel, infinitely rising trajectories. The equivalence between the partial-wave Bethe-Salpeter equation and the continuous-dimensional formalism used here is explicitly developed. Suggestions are made for extending the method to Bethe-Salpeter equations involving spin or multichannel effects.