  Parametric representations of Feynman integrals have a key property: many,
frequently all, of the Landau singularities appear as endpoint divergences.
This leads to a geometric interpretation of the singularities as faces of
Newton polytopes, which facilitates algorithmic evaluation by sector
decomposition and asymptotic expansion by the method of regions. Here we
identify cases where some singularities appear instead as pinches in parametric
space for general kinematics, and we then extend the applicability of sector
decomposition and the method of regions algorithms to such integrals, by
dissecting the Newton polytope on the singular locus. We focus on $2\to 2$
massless scattering, where we show that pinches in parameter space occur
starting from three loops in particular nonplanar graphs due to cancellation
between terms of opposite sign in the second Symanzik polynomial. While the
affected integrals cannot be evaluated by standard sector decomposition, we
show how they can be computed by first linearising the graph polynomial and
then splitting the integration domain at the singularity, so as to turn it into
an endpoint divergence. Furthermore, we demonstrate that obtaining the correct
asymptotic expansion of such integrals by the method of regions requires the
introduction of new regions, which can be systematically identified as facets
of the dissected polytope. In certain instances, these hidden regions
exclusively govern the leading power behaviour of the integral. In momentum
space, we find that in the on-shell expansion for wide-angle scattering the new
regions are characterised by having two or more connected hard subgraphs, while
in the Regge limit they are characterised by Glauber modes.
