The purpose of the present invention is to provide a cap and a container with a cap with which flap reversal during cap-opening is reliably prevented while keeping the user from feeling that torque during cap-opening is inappropriate. Provided is a cap (10) provided with a TE band (30) that is joined to the lower edge of a cap body (20) via a weakened section (40), wherein: the TE band (30) has aflap (32) that is folded back toward the inside of a band body section (31); the flap (32) has a flange-engaging section (32b) capable of engaging with the lower side of a container flange (53) when fitted on the container, and an orientation-maintaining piece (32c) disposed on the outside of the container flange (53) when fitted on the container; and on the inner circumferential wall of the bandbody section (31), a clearance area (22c), which forms a gap between the outer circumferential wall of the orientation-maintaining piece (32c) and the inner circumferential wall of the band body section (31), is formed.