Why are diagonal arrows disallowed in Jablonski diagrams?
Since nobody has responded to this yet, I'll offer a short answer. 

In a Jablonski diagram, the electronic states are grouped together horizontally by spin multiplicity $2S+1$. So, in the simplest example, you see a column of singlet states $S=0$ and a column of triplet states $S=1$. (Quite often, the vibrational levels are also illustrated schematically). Diagonal lines, then, represent transitions between states of different multiplicity. 

These are more usually termed "forbidden transitions" rather than "disallowed". An example is phosphorescence, the decay from an excited triplet to a lower-lying singlet, most commonly the ground state. There is a general selection rule, $\Delta S=0$, which in principle makes the transition moment vanish. However, the rule can be violated: one possible reason is that $S$ is not a perfectly good quantum number due to spin-orbit coupling. Typically, the consequence is that the transition occurs very infrequently (which is why phosphorescence occurs slowly, and weakly).

So, it is not "disallowed" to put diagonal lines on a Jablonski diagram,
but in most cases they correspond to weak, "forbidden", transitions.