  We present a general procedure, based on the Ehlers transformation of the
Ernst equations, to add the gravitomagnetic mass to the whole
Plebanski-Demianski family of solutions. We can efficiently generate a large
class of accelerating black holes, such as Reissner-Nordstrom or Kerr-Newman,
endowed with the NUT parameter. The full rotating version carries a couple of
independent NUT charges, one associated to the black hole and the other to the
accelerating Rindler background. The two NUT parameters can be coupled to
remove the axial irregularity which causes the Misner string, still remaining
with a Lorentzian spacetime, without the need to impose periodic time. All the
metrics we build are not of D-type according to the Petrov classification, but
type-I, so they belong to a more general category with respect to C-metrics and
the Plebanski-Demianski seed. A convenient form of the most general type D
black hole solution in general relativity, coupled with Maxwell
electromagnetism, is obtained when switching off one of the two NUT parameters.
