Black holes of type D revisited: relating their various metric forms

We investigate a complete family of spacetimes which represent black holes with rotation, NUT twist, acceleration, electric and magnetic charges. These are exact solutions of the Einstein-Maxwell equations with any cosmological constant, such that the (non-null) electromagnetic field is aligned with both the double-degenerate principal null directions of the Weyl tensor. In particular, we explicitly relate various coordinates and the corresponding physical parameters of such solutions, namely the original Plebanski-Demianski (PD) form, the convenient Astorino (A) form which was found recently, and formally improved here (A+), the Griffiths-Podolsky (GP), and Podolsky-Vratny (PV) form of the metric. It is demonstrated that, if properly mapped and physically interpreted, all these representations cover the complete class of type D black holes. Using the new A-parameters, the two main PD quartic metric functions are factorized into the product of quadratic expressions, enabling thus an explicit analysis. Moreover, we clarify the role of the twist parameter $\omega$, related to both the Kerr-like rotation and the NUT parameters $a$ and $l$, respectively. Special attention is payed to the elusive subclass of accelerating NUT black holes with $a=0$.