  The Einstein equivalence principle in general relativity allows us to
interpret accelerating black holes as a black hole immersed into the
gravitational field of a larger companion black hole. Indeed it is demonstrated
that C-metrics can be obtained as a limit of a binary system where one of the
black holes grows indefinitely large, becoming a Rindler horizon. When the
bigger black hole, before the limiting process, is of Schwarzschild type we
recover usual accelerating black holes belonging to the Plebanski-Demianski
class, thus type D. Whether the greater black hole carries some extra features,
such as electric charges or rotations, we get generalised accelerating black
holes which belong to a more general class, the type I. In that case the
background has a richer structure, reminiscent of the physical features of the
inflated companion, with respect to the standard Rindler spacetime. This
insight allows us to build a general type D metric, describing an accelerating
Kerr-NUT black hole. It has well defined limits to all the type-D black holes
of general relativity, including the elusive (type-D) accelerating Taub-NUT
spacetime. Extension to the presence of the cosmological constant is also
provided.
