  In this manuscript we report the flag-dipole spinors dual structure direct
definition and analyze the properties behind the corresponding operator which
generates such structure. This particular construction may be interesting for
cosmological, phenomenological and mathematical physics applications. In
addition, we analyse the behaviour of the flag-dipole spinors under action of
discrete symmetries, facing an \emph{unconventional} property encoded on
$(\mathcal{CPT})^2$.
