<p>The Talbot effect describes the periodic revivals of field patterns, and is ubiquitous across wave systems. In optics, it is mostly known for its manifestations in space and time, but it is also observed in the wavevector and frequency spectra owing to the Fourier duality. Recently, the Talbot self-imaging has been shown separately in the azimuthal angle and orbital angular momentum (OAM) domains. Here we reveal the missing link between them and demonstrate the generalized angle–OAM Talbot effect. Versatile transformations of petal fields and OAM spectra are experimentally demonstrated, based on the synergy of angular Talbot phase modulation and light propagation in a ring-core fibre. Moreover, the generalized self-imaging concept leads to new realizations in mode sorting, which separate OAM modes in a modulo manner, theoretically free from any crosstalk within the congruence classes of OAM modes. We design and experimentally construct various mode sorters with excellent performance, and show the unconventional behaviour of Talbot-based sorters where neighbouring OAM modes can be mapped to positions that are far apart. Besides its fundamental interest, our work has applications in OAM-based information processing, and implies that the physical phenomena in time–frequency and angle–OAM domains are broadly connected and that their processing techniques may be borrowed interchangeably.</p>