  We scrutinize the many known forms of BRST symmetries, as well as some new
ones, realized within a prototypical first-class system. Similarities and
differences among ordinary BRST, anti-BRST, dual-BRST and anti-dual-BRST
symmetries are highlighted and discussed. We identify a precise
$\mathbb{Z}_4\times\mathbb{Z}_2$ discrete group of symmetries of the ghost
sector, responsible for connecting the various forms of BRST transformations.
Considering a Hamiltonian approach, those symmetries can be interrelated by
canonical transformations among ghost variables. However, the distinguished
characteristic role of the dual BRST symmetries can be fully appreciated within
a gauge-fixed Lagrangian viewpoint. New forms of BRST symmetries are given, a
set generalizing particular ones previously reported in the literature as well
as a brand new unprecedented set. The featured gauge invariant prototypical
first-class system encompasses an extensive class of physical models and sheds
light on previous controversies in the current quantum field theory literature.
