The upper-ocean relative vorticity has been found to be cyclonically skewed, but altimetry observations indicate that long‐lifespan mesoscale eddies tend to be anticyclonic. We are thus interested in whether cyclonic or anticyclonic eddies are more unstable under similar circumstances. Here we use submesoscale‐resolving simulations of idealized mesoscale eddies, incorporating theoretical analyses, to investigate asymmetries of submesoscale instabilities within the anticyclones and cyclones. It is found that submesoscale filaments initiate at regions with the largest horizontal buoyancy gradients for both anticyclones and cyclones, but these filaments subsequently rotate outward in anticyclones while inward in cyclones. Hence submesoscales are more vigorous at anticyclone peripheries and the cyclone center. Such differing distributions and evolutions of submesoscale processes are primarily caused by changes in the background stratification associated with the decaying of mesoscale eddies. The active submesoscales near the cyclone center eventually distort its core structure radically, whereas the anticyclone remains largely unaffected.