  We search for a \Lg\ interpretation of non-diagonal modular invariants of
tensor products of minimal $n=2$ superconformal models, looking in particular
at automorphism invariants and at some exceptional cases. For the former we
find a simple description as \lgo s, which reproduce the correct chiral rings
as well as the spectra of various Gepner--type models and orbifolds thereof. On
the other hand, we are able to prove for one of the exceptional cases that this
conformal field theory cannot be described by an orbifold of a \Lg\ model with
respect to a manifest linear symmetry of its potential.
