We propose that several short-duration events observed in past stellar occultations by Chiron were produced by ring material. Some similarities between these events and the characteristics of Chariklo's rings could indicate common mechanisms around centaurs. From a reanalysis of the stellar occultation data in the literature, we determined two possible orientations of the pole of Chiron's rings, with ecliptic coordinates λ = (352 ± 10)°, β = (37 ± 10)° or λ = (144 ± 10)°, β = (24 ± 10)°. The mean radius of the rings is (324 ± 10) km. One can use the rotational lightcurve amplitude of Chiron at different epochs to distinguish between the two solutions for the pole. Both solutions imply a lower lightcurve amplitude in 2013 than in 1988, when the rotational lightcurve was first determined. We derived Chiron's rotational lightcurve in 2013 from observations at the 1.23 m CAHA telescope, and indeed its amplitude was smaller than in 1988. We also present a rotational lightcurve in 2000 from images taken at the CASLEO 2.15 m telescope that is consistent with our predictions. Out of the two poles, the λ = (144 ± 10)°, β = (24 ± 10)° solution provides a better match to a compilation of rotational lightcurve amplitudes from the literature and those presented here. We also show that using this preferred pole orientation, Chiron's long-term brightness variations are compatible with a simple model that incorporates the changing brightness of the rings while the tilt angle with respect to the Earth is changing with time. Also, the variability of the water ice band in Chiron's spectra as seen in the literature can be explained to a large degree by an icy ring system whose tilt angle changes with time and whose composition includes water ice, analogously to the case of Chariklo. We present several possible formation scenarios for the rings from qualitative points of view and speculate on why rings might be common in centaurs. We also speculate on whether the known bimodal color distribution of the centaurs could be due to centaurs with rings and centaurs without rings. Table 1 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/576/A18