  We explore various aspects of the self-dual Pleba\'nski-Demia\'nski family in
the Euclidean Einstein-Maxwell-$\Lambda$ system. The Killing-Yano tensor which
was recently found by Yasui and one of the present authors allows us to prove
that the self-dual Pleba\'nski-Demia\'nski metric can be brought into the
self-dual Carter metric by an orientation-reversing coordinate transformation.
We show that the self-dual Pleba\'nski-Demia\'nski solution admits two
independent Killing spinors in the framework of $N=2$ minimal gauged
supergravity, whereas the non-self-dual solution admits only a single Killing
spinor. This can be demonstrated by casting the self-dual
Pleba\'nski-Demia\'nski metric into two distinct Przanowski-Tod forms. As a
by-product, a new example of the three-dimensional Einstein-Weyl space is
presented. We also prove that the self-dual Pleba\'nski-Demia\'nski metric
falls into two different Calderbank-Pedersen families, which are determined by
a single function subjected to a linear equation on the two dimensional
hyperbolic space. Furthermore, we consider the hyper-K\"ahler case for which
the metric falls into the Gibbons-Hawking class. We find that the condition for
the nonexistence of Dirac-Misner string enforces the solution with a
nonvanishing acceleration parameter to the Eguchi-Hanson space.
