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Aims. 2015 BZ509 is the first asteroid confirmed to be in retrograde co-orbitresonance (or 1/-1 resonance) with the giant planets in the solar system. WhileSaturn is the only giant planet whose Trojans are not discovered until now, weidentify some small bodies among Centaurs and Damocloids that are potentiallyin 1/-1 resonance with Saturn in the present study. Methods. We integratenumerically the motion of the 1000 clones (include the nominal orbit) of eachCentaur whose orbit has a semi-major axis between 9.3 au and 9.8 au and aninclination i > 90 deg. To confirm and evaluate the 1/-1 resonantconfigurations mentioned above, we introduce a useful one-degree integrableapproximation for planar 1/-1 resonance. Results. We identify four candidatespotentially in 1/-1 resonance with Saturn. The capture in this particularresonant state during the 40000 yr integration timespan is very common for 2006RJ2 (906/1000 clones), 2006 BZ8 (878/1000 clones), and 2017 SV13 (998/1000clones), and it is less likely for 2012 YE8 (426/1000 clones). According to ourstatistical results, 2006 RJ2 is the best candidate to be currently in a 1/-1mean motion resonance with Saturn, and 2017 SV13 is another important potentialcandidate. Moreover, 2012 YE8 and 2006 BZ8 are also Centaurs of interest buttheir current and long-term 1/-1 resonant state with Saturn is less likely. Theproportions of the clones captured in the relative long-term stable co-orbitresonance (over 10000 yr) are also given. Conclusions. Small bodies inretrograde co-orbit resonance with giant planets may be more common thanpreviously expected. Identification of these potential mysterious minor bodiesencourages the search for such objects on a larger scale in our solar system.The findings of this paper are also useful for understanding the origin anddynamical evolution of the Centaurs and Damocloids on retrograde orbits.

Centaurs potentially in retrograde co-orbit resonance with Saturn