Optical vortices (OVs) promise to greatly enhance optical information capacity via orbital angular momentum multiplexing. The need for the on-chip integration of orbital angular momentum technologies has prompted research into subwavelength-confined polaritonic OVs. However, the topological order imprinted by the structure used for transduction from free-space beams to surface polaritons is inherently fixed after fabrication. Here we overcome this limitation via dispersion-driven topological charge multiplication. We switch the OV topological charge within a small frequency range (~3%) by leveraging the strong sublinear dispersion of low-loss surface phonon polaritons on silicon carbide membranes. Applying the Huygens principle, we quantitatively evaluate the topological order of experimental OVs detected by near-field imaging. We further explore the deuterogenic effect, which predicts the coexistence of multiple topological charges in higher-order polaritonic OVs. Our work demonstrates a viable method to manipulate the topological charge of polaritonic OVs, paving the way for the exploration of novel orbital-angular-momentum-enabled light–matter interactions at mid-infrared frequencies.