  In this paper, we study the behaviour of TF-isomorphisms, a natural
generalisation of isomorphisms. TF-isomorphisms allow us to simplify the
approach to seemingly unrelated problems. In particular, we mention the
Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and
Stability of Graphs. We start with a study of invariance under TF-isomorphisms.
In particular, we show that alternating trails and incidence double covers are
conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms
between graphs or digraphs. We then define an equivalence relation and
subsequently relate its equivalence classes to the incidence double cover of a
graph. By directing the edges of an incidence double cover from one colour
class to the other and discarding isolated vertices we obtain an invariant
under TF-isomorphisms which gathers a number of invariants. This can be used to
study TF-orbitals, an analogous generalisation of the orbitals of a permutation
group.
