I wondered if there is someone who knows an example as described in the title.
If $M$ is a closed oriented smooth 4-manifold, then a homology orientation on $M$ is a choice of orientations on $H_1(M;\mathbb{R})\oplus H_2(M;\mathbb{R})$.
The definition of an (usual) orientation is well-known.
So are there diffeomorphisms $\varphi:M\to M$ which change one but leaves the other invariant?
In the case of simply connected closed oriented smooth 4-manifolds the choice of an homology orientation is equivalent to the choice of an orientation of $H_2(M;\mathbb{R})$. Is there even an example for simply connected manifolds?
Consider $M=S^1\times S^3$, and $f_1$ a diffeomorphism of $S^1$ which does not preserve the orientation, $f_2$ an automorphism of $S^3$ which does not preserve the orientation $(f_1,f_2)$ does not preserve the homology orientation, but preserves the orientation of $M$.