  Three different hybrid Vlasov-fluid systems are derived by applying reduction
by symmetry to Hamilton's variational principle. In particular, the discussion
focuses on the Euler-Poincar\'e formulation of three major hybrid MHD models,
which are compared in the same framework. These are the current-coupling scheme
and two different variants of the pressure-coupling scheme. The Kelvin-Noether
theorem is presented explicitly for each scheme, together with the Poincar\'e
invariants for its hot particle trajectories. Extensions of Ertel's relation
for the potential vorticity and for its gradient are also found in each case,
as well as new expressions of cross helicity invariants.
