Explicit solutions of the equations of Kirchhoff’s theory of elastic rods are employed to derive properties of the tertiary structure of a looped segment of DNA that is subject to geometric constraints imposed at its end points by bound proteins. In appropriate circumstances small changes in such boundary data cause a nearly planar loop to undergo a continuous and reversible transition that can be described as a 180 degrees rotation taking the loop from an uncrossed to a singly crossed structure in which sequentially separated base pairs are brought into proximity. Expressions are derived relating points and angles of crossing to end conditions, and results are presented that facilitate the calculation of changes in elastic energy during such transitions.