  We introduce a class of surfaces in euclidean space motivated by a problem
posed by \'{E}lie Cartan. This class furnishes what seems to be the first
examples of pairs of non-congruent surfaces in euclidean space such that, under
a diffeomorphism $\Phi$, lines of curvatures are preserved and principal
curvatures are switched. We show how to construct such surfaces using
holomorphic data and discuss their relation with minimal surfaces. We also
prove that if the diffeomorphism $\Phi$ preserves the conformal class of the
third fundamental form, then all examples belong to the class of surfaces that
we deal with in this work.
