  We show that two $\operatorname{Pin}$-structures on a surface differ by a
diffeomorphism of the surface if and only if they are cobordant (for
comparison, the analogous fact has already been shown for
$\operatorname{Spin}$-structures). We give a construction that shows that this
does not extend to dimensions greater than two. In addition, we count the
number of $\operatorname{Pin}$-structures on a surface in a given cobordism
class.
