Manifold with an orientation-reversing self-homeomorphism

Definition
A connected orientable manifold is termed a manifold with an orientation-reversing self-homeomorphism if there is a homeomorphism from the manifold to itself that, if applied to any orientation of the manifold, produces the opposite orientation.

For a compact connected orientable manifold
For a compact connected orientable manifold, a self-homeomorphism is orientation-reversing if it induces the multiplication by $$-1$$ map on the top homology, or equivalently, sends a fundamental class to its negative. A manifold with an orientation-reversing self-homeomorphism is thus a manifold for which there exists a self-homeomorphism with such an effect.

Examples
Below are some examples of compact connected orientable manifolds and whether or not they have orientation-reversing self-homeomorphisms:

Facts

 * Given two connected orientable manifolds $$M_1$$ and $$M_2$$, if either of them possesses an orientation-reversing self-homeomorphism, so does the product $$M_1 \times M_2$$. The trick is to take the Cartesian product of the orientation-reversing self-homeomorphism on one manifold and the identity map on the other.