  Impulsive gravitational waves in Minkowski space were introduced by Roger
Penrose at the end of the 1960s, and have been widely studied over the decades.
Here we focus on non-expanding waves which later have been generalised to
impulses travelling in all constant-curvature backgrounds, that is also the
(anti-)de Sitter universe. While Penrose's original construction was based on
his vivid geometric `scissors-and-paste' approach in a flat background, until
now a comparably powerful visualisation and understanding have been missing in
the ${\Lambda\not=0}$ case. In this work we provide such a picture: The
(anti-)de Sitter hyperboloid is cut along the null wave surface, and the
`halves' are then re-attached with a suitable shift of their null generators
across the wave surface. This special family of global null geodesics defines
an appropriate comoving coordinate system, leading to the continuous form of
the metric. Moreover, it provides a complete understanding of the nature of the
Penrose junction conditions and their specific form. These findings shed light
on recent discussions of the memory effect in impulsive waves.
