In consumer-market containers for beverages or other contents under pressure, the cap provides a gas-tight seal by way of two mechanisms that work interdependently. The first is a pressure grip configured to grip the outside wall of the container. The second is twist threads, or some other second attachment structure, that urges the inside of the cap (or, typically, a cap liner disposed thereon) against the rim of the container. Release of pressurized gases when the container is first opened can be by way of release vents in the cap, or via the bottom of the cap through the twist thread area. In the latter case, particular configurations of the pressure grip are relied on to establish the gas escape path.