The on-momentum description of linear coupling between horizontal and vertical betatron motion is extended to include off-momentum particles, introducing a vector quantity called the ``skew chromaticity``. This vector tends to be long in large superconducting storage rings, where it restricts the available working space in the tune plane, and modifies collective effect stability criteria. Skew chromaticity measurements at the Cornell Electron Storage Ring (CESR) and at the Fermilab Tevatron are reported, as well as tracking results from the Relativistic Heavy Ion Collider (RHIC). The observation of anomalous head-tail beam Iowa new the tune diagonal in the Tevatron are explained in terms of the extended theory, including modified criteria for headtail stability. These results are confirmed in head-tail simulations. Sources of skew chromaticity are investigated
