  Common intuition in physics is based on the concept of orthogonal eigenmodes.
Those are well de- fined solutions of Hermitian equations used to describe many
physical situations, from quantum mechanics to acoustics. A large variety of
non-Hermitian problems, including gravitational waves close to black holes or
leaky electromagnetic cavities require the use of bi-orthogonal eigenbasis.
Physical consequences of non-Hermiticity challenge our physical understanding.
However, the usual need to compensate for energy losses made the few successful
attempts to probe non-Hermiticity extremely complicated. We show that this
issue can be overcome considering localized plasmonic systems. Indeed, since
the non-Hermiticity in these systems does not stem from from temporal
invariance breaking but from spatial symmetry breaking, its consequences can be
observed more easily. We report on the theoretical and experimental evidence of
non-Hermiticity induced strong coupling between surface plasmon modes of
different orders within silver nano-crosses. The sym- metry conditions for
triggering this counter-intuitive self-hybridization phenomenon are provided.
Similarly, observable effects are expected to be observed in any system
subtending bi-orthogonal eigenmodes.
