The Kerr-NUT black hole is not just a cosmic vacuum cleaner. It's a solution to Einstein's equations where the hole has mass, rotation, and a strange NUT parameter. This parameter behaves like a magnetic charge for gravity: seemingly invisible, yet it alters the properties of surrounding space.
The authors applied to this hole the thermodynamic approach that worked for ordinary black holes. And they found a new conserved quantity: it is dual to angular momentum. It's like a mirror reflection: angular momentum accounts for rotation, while its twin accounts for something like "hidden rotation" associated with the NUT charge. The quantity always exists, but in thermodynamics it shows up only when the hole actually spins (parameter $a\neq 0$).
The resulting picture is beautiful: together with the two charges ($N_c = cn$ and $n$), all thermodynamic relations (the first law, the Smarr formula, and the Gibbs–Duhem equation) hold flawlessly. Nature seems to hint that every important quantity has its own invisible partner.
The NUT charge can be imagined as a "gravitational magnet"—it doesn't attract iron, but it makes test particles move along unusual trajectories, as if space itself nudges them sideways.