Nonorientable Lagrangian cobordisms between Legendrian knots

In the symplectization of standard contact 33-space, R×R3ℝℝ3, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus 00. We show that any Legendrian knot has a nonorientable Lagrangian endocobordism, and that the cross-cap genus of such a nonorientable Lagrangian endocobordism must be a positive multiple of 44. The more restrictive exact, nonorientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do exist for any stabilized Legendrian knot. Moreover, the relation defined by exact, nonorientable Lagrangian cobordism on the set of stabilized Legendrian knots is symmetric and defines an equivalence relation, a contrast to the nonsymmetric relation defined by orientable Lagrangian cobordisms