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.)

1. 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.
2. In any octahedron, the Mayer-Vietoris triangles are exact (Lemma 3.6, asserted to be a "lengthy but elementary diagram chase").
3. 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).
4. 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?

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.)

1. 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.
2. In any octahedron, the Mayer-Vietoris triangles are exact (Lemma 3.6, asserted to be a "lengthy but elementary diagram chase").
3. 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).
4. 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?