The author deals with a split-and-merge transformation of interval partitions which combines features of a model studied by A. Gnedin and S. Kerov [Comb. Probab. Comput. 10, No. 3, 213-217 (2001; Zbl 0976.62008) and Random Struct. Algorithms 20, No. 1, 71-88 (2002; Zbl 1007.05012)] and of another one studied by N. V. Tsilevich [Theory Probab. Appl. 44, No. 1, 60-74 (1999); translation from Teor. Veroyatn. Primen. 44, No. 1, 55-73 (1999; Zbl 0960.60012)] and E. Mayer-Wolf, O. Zeitouni, and M. Zerner [Electron J. Probab. 7, Paper No. 8 (2002; Zbl 1007.60100)]. The interval partition generated by a suitable Poisson process is invariant for the model considered, which is a Markovian one. This provides a simple proof of a result in the last reference above, that a Poisson-Dirichlet distribution is invariant for a closely related fragmentation-coagulation process. The author proves uniqueness and convergence to the invariant measure for the split-and-merge transformation of interval partitions while the corresponding problems for the fragmentation-coagulation process remain open. Uniqueness in the latter case is shown to have some connection with Kingman’s theory of exchangeable random partitions [see J. F. C. Kingman, Stochastic Processes Appl. 13, 235-248 (1982; Zbl 0491.60076)].