In May's paper The additivity of traces in triangulated categories he has the following things to say about this property. (At least, I think it is an equivalent property -- his notation is different and some minus signs are in different places.)
- Call a triangle exact if it induces long exact sequences upon application of both representable and corepresentable functors (Definition 3.2). All distinguished triangles are exact, but not conversely.
- In any octahedron, the Mayer-Vietoris triangles are exact (Lemma 3.6, asserted to be a "lengthy but elementary diagram chase").
- In any triangulated category arising from a well-behaved Quillen model category, given the rest of an octahedron there is some choice of $x$ and $y$ making the Mayer-Vietoris triangles distinguished (section 5). He proposes to call a triangulated category with this property strong (Definition 3.8).
- In general, the Mayer-Vietoris triangles are not distinguished for all choices of $x$ and $y$. He sketches an argument for this that I don't completely follow, citing "Faisceaux pervers" by Beilinson, Bernstein, and Deligne (Remark 3.7).
The last point seems to contradict what you say Iversen shows. I don't have access to Iversen; does he actually show this for all choices of $x$ and $y$, or only for the particular ones constructed in his proof of the octahedral axiom?