A circular stapler apparatus for stapling tissue has a circular stapling head operable to drive staples toward an anvil to form the staples in a circular array. The anvil includes an anvil head, a proximal shaft extending proximally from the anvil head and having a proximal end disposed in a first plane, and a first pivot connecting the anvil head to the proximal shaft. The first pivot is operable to be disposed in a second plane that is laterally offset from the first plane. The anvil is configured to rotate via movement through multiple pivot points. The anvil may be configured to rotate to a position in which a portion of the anvil head is disposed below the first pivot and between the first pivot and the proximal shaft such that the anvil head is acutely angled with respect to a longitudinal axis of the proximal shaft.