In May's paper [The additivity of traces in triangulated categories](https://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf) 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.
1. In any octahedron, the Mayer-Vietoris triangles are exact (Lemma 3.6, asserted to be a "lengthy but elementary diagram chase").
1. 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).
1. 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?