  Four observations compose the main results of this note. The first records
the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times
S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is
homeomorphic but not diffeomorphic to the total space of the non-trivial
2-sphere bundle over the real projective plane $S(2\gamma \oplus \mathbb{R})$.
The second observation is that there is a 5-dimensional cobordism with a single
2-handle between the 4-manifold $Y$ and a mapping torus that was used by
Cappell-Shaneson to construct an exotic $\mathbb{R} P^4$. This construction of
$Y$ is similar to the one of the Cappell-Shaneson homotopy 4-spheres. The third
observation is that twisting an embedded real projective plane inside $Y$
produces a manifold that is homeomorphic but not diffeomorphic to the circle
sum of two copies of $\mathbb{R}P^4$. Knotting phenomena of 2-spheres in
non-orientable 4-manifolds that stands in glaring contrast with known phenomena
in the orientable domain is pointed out in the fourth observation.
