  The scattering structures in the ISM responsible for so-called ``extreme
scattering events" (ESEs), observed in quasars and pulsars, remain enigmatic.
Current models struggle to explain the high-frequency light curves of ESEs, and
a recent analysis of a double lensing event in PSR\,B0834+06 reveals features
of ESEs that may also be challenging to accommodate via existing models. We
propose that these features arise naturally when the lens has a cusp-like
profile, described by the elementary $A_3$ cusp catastrophe. This is an
extension of previous work describing pulsar scintillation as arising from
$A_2$ fold catastrophes in thin, corrugated plasma sheets along the line of
sight. We call this framework of describing the lens potentials via elementary
catastrophes ``doubly catastrophic lensing", as catastrophes (e.g. folds and
cusps) have long been used to describe universal features in the light curves
of lensing events that generically manifest, regardless of the precise details
of the lens. Here, we argue that the lenses themselves may be described by
these same elementary structures. If correct, the doubly catastrophic lensing
framework would provide a unified description of scintillation and ESEs, where
the lenses responsible for these scattering phenomena are universal and can be
fully described by a small number of unfolding parameters. This could enable
their application as giant cosmic lenses for precision measurements of coherent
sources, including FRBs and pulsars.
