  The object of study are almost complex manifolds with a pair of Norden
metrics, mutually associated by means of the almost complex structure. More
precisely, a torsion-free connection and tensors with geometric interpretation
are found which are invariant under the twin interchange, i.e. the swap of the
counterparts of the pair of Norden metrics and the corresponding Levi-Civita
connections. A Lie group depending on four real parameters is considered as an
example of a 4-dimensional manifold of the studied type. The mentioned
invariant objects are found in an explicit form.
