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 ω, 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 $a=0$.