  We study a complex intertwining relation of second order for Schroedinger
operators and construct third order symmetry operators for them. A modification
of this approach leads to a higher order shape invariance. We analyze with
particular attention irreducible second order Darboux transformations which
together with the first order act as building blocks. For the third order
shape-invariance irreducible Darboux transformations entail only one sequence
of equidistant levels while for the reducible case the structure consists of up
to three infinite sequences of equidistant levels and, in some cases, singlets
or doublets of isolated levels.
