The Circles in Matched are Cassia's grandmother's compact her grandfather gave her (that he hid poems in), Ky's Compass that isn't registered with the Society that he brought from the Outer Provinces and wanted Cassia to borrow, and Bram's silver watch that Bram and Cassia's Grandfather gave him.
Circles represent wholeness and unity, and I think they are all good symbols for those.
