  The way we organise perturbation theory is of fundamental importance both for
computing the observables of relevance and for extracting fundamental physics
out of them. If on one hand the different ways in which the perturbative
observables can be written make manifest different features (e.g. symmetries as
well as principles such as unitarity, causality and locality), on the other
hand precisely demanding that some concrete features are manifest lead to
different ways of organising perturbation theory. In the context of flat-space
scattering amplitudes, a number of them are already known and exploited, while
much less is known for cosmological observables. In the present work, we show
how to systematically write down both the wavefunction of the universe and the
flat-space scattering amplitudes, in such a way that they manifestly show
physical poles only. We make use of the invariant definition of such
observables in terms of {\it cosmological polytopes} and their {\it scattering
facet}. In particular, we show that such representations correspond to
triangulations of such objects through hyperplanes identified by the
intersection of their facets outside of them. All possible triangulations of
this type generate the different representations. This allows us to provide a
general proof for the conjectured all-loop causal representation of scattering
amplitudes. Importantly, all such representations can be viewed as making
explicit a subset of compatible singularities, and our construction provides a
way to extend Steinmann relations to higher codimension singularities for both
the flat-space scattering amplitudes and the cosmological wavefunction.
