# Contact manifolds that are not cooriented

One always sees the definition of a contact manifold $(X,\xi)$ as an odd dimensional manifold with a hyperplane distribution $\xi$ which can locally be expressed as $\xi = \ker \alpha$ for a 1-form $\alpha$. But in fact, it seems that in every example I know of, one always assumes that $\xi$ is cooriented, and hence we can write $\xi = \ker \alpha$ globally.

Is there a reason (other than historical) as to why coorientation wasn't built in automatically in the definition of a contact manifold? It seems strange that this isn't required in the definition.

the space of contact elements (the grassmanian of hyperplane) on a manifold is isomorphic $P(T^*M)=T^*M\setminus M_0/\mathbb{R}$ . The local contact forms are locally given by the Liouville form on a local section of the quotient map by the action of $\mathbb{R}$. However you cannot define a global form (you can on a double cover which corresponds to the space cooriented contact elements).
This space is of interest, for instance submanifolds of $M$ give Legendrian submanifolds of $P(T^*M)$, and isotopy give Legendrian isotopy. Section of $P(T^*M)$ which are contact submanifolds are exactly the contact structure on $M$, etc.