The rotation period of a gas giant's magnetic field (called the System III reference frame) is commonly used to infer its bulk rotation1. Saturn's dipole magnetic field is not tilted relative to its rotation axis (unlike Jupiter, Uranus and Neptune), so the surrogate measure of its long-wavelength (kilometric) radiation is currently used to fix the System III rotation period2. The period as measured now by the Cassini spacecraft is up to 7 min longer3 than the value of 10 h 39 min 24 s measured 28 years ago by Voyager2. Here we report a determination of Saturn's rotation period based on an analysis of potential vorticity. The resulting reference frame (which we call System IIIw) rotates with a period of 10 h 34 min 13 20 s. This shifted reference frame is consistent with a pattern of alternating jets on Saturn that is more symmetrical between eastward and westward flow. This suggests that Saturn's winds are much more like those of Jupiter than hitherto believed4.