  We present and analyze a class of exact spacetimes which describe
accelerating black holes with a NUT parameter. First, we verify that the
intricate metric found by Chng, Mann and Stelea in 2006 indeed solves
Einstein's vacuum field equations of General Relativity. We explicitly
calculate all components of the Weyl tensor and determine its algebraic
structure. As it turns out, it is actually of algebraically general type I with
four distinct principal null directions. It explains why this class of
solutions has not been (and could not be) found within the large
Plebanski-Demianski family of type D spacetimes. Then we transform the solution
into a much more convenient metric form which explicitly depends on three
physical parameters: mass, acceleration, and the NUT parameter. These
parameters can independently be set to zero, recovering thus the well-known
spacetimes in standard coordinates, namely the C-metric, the Taub-NUT metric,
the Schwarzschild metric, and flat Minkowski space. Using this new metric, we
investigate physical and geometrical properties of such accelerating NUT black
holes. In particular, we localize and study four Killing horizons (two
black-hole plus two acceleration) and investigate the curvature. Employing the
scalar invariants we prove that there are no curvature singularities whenever
the NUT parameter is nonzero. We identify asymptotically flat regions and
relate them to conformal infinities. This leads to a complete understanding of
the global structure. The boost-rotation metric form reveals that there is
actually a pair of such black holes. They uniformly accelerate in opposite
directions due to the action of rotating cosmic strings or struts located along
the corresponding two axes. Rotation of these sources is directly related to
the NUT parameter. In their vicinity there are pathological regions with closed
timelike curves.
