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Möbius-ring resonators stem from a well-studied and fascinating geometrical structure that features a one-sided topology; the Möbius strip, and have been shown to exhibit fermion rotational symmetry with respect to a ring resonator with no twist (which exhibits boson rotational symmetry) (see PhysRevLett.101.247701). Here, we present a new type of resonator through the formation of twisted hollow structures using equilateral triangular cross-sections, which leads to the realization of a cavity with anyon rotational symmetry. Unlike all previous cavity resonators, the anyon resonator permits the existence of bulk resonant modes that exhibit non-zero electromagnetic helicity in vacuo, with a non-zero overlap of the electric and magnetic mode eigenvectors, $\int \mathbf{E}_p\cdot\mathbf{B}_p~dτ$, integrated over the cavity volume. In the upconversion limit, we show that these non-zero helical modes couple naturally to ultra-light dark matter axions within the bandwidth of the resonator by adding amplitude-modulated sidebands through the axion-photon chiral anomaly. Thus, we show a sensitive ultra-light dark matter experiment may be realized by implementing such a resonator in an ultra-stable oscillator configuration and searching for signals in the Fourier spectrum of amplitude fluctuations. This removes the typical requirement for an external magnetic field and therefore permits the use of superconducting materials to reduce surface losses and enhance sensitivity to axions.