  In the symplectization of standard contact $3$-space, $\mathbb R \times
\mathbb R^3$, 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 $0$. We show that any
Legendrian knot has a non-orientable Lagrangian endocobordism, and that the
crosscap genus of such a non-orientable Lagrangian endocobordism must be a
positive multiple of $4$. The more restrictive exact, non-orientable 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, non-orientable Lagrangian cobordism on the set of stabilized Legendrian
knots is symmetric and defines an equivalence relation, a contrast to the
non-symmetric relation defined by orientable Lagrangian cobordisms.
