Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the axioms. For example, the octahedron axiom gives an exact triangle $E\to E^{(2)}\to E\xrightarrow[]{\Sigma k\circ j}\Sigma E$, where $E^{(2)}$ is from the exact triangle $D\xrightarrow[]{\varphi\circ\varphi}D\xrightarrow[]{}E^{(2)}\to\Sigma D$. All this resembles a beginning of a spectral sequence obtained from an exact couple (e. g. obviously the composite $d:E\xrightarrow[]j\Sigma D\xrightarrow[]{\Sigma k}\Sigma E$ satisfies $\Sigma d\circ d=0$) but it quickly becomes too entangled for me.

Can anyone give me a reference for analyzing such situations? At least in simpler situations when e. g. $\varphi\circ\varphi$ is $\varphi$ or zero or invertible? Or on the other hand in more general situations when $\varphi$ has nonzero degree (i. e. $\varphi:\Sigma^nD\to D$ for any $n\in\mathbb Z$)?

I have vague feeling that something like this takes place in connection with the Devinatz-Hopkins-Smith theorem, and that Bondal-Kapranov's twisted complexes might be relevant, but cannot come up with anything definite...