  We provide a Hamiltonian analysis of the Mixmaster Universe dynamics showing
the covariant nature of its chaotic behavior with respect to any choice of time
variable. We construct the appropriate invariant measure for the system (which
relies on the existence of an ``energy-like'' constant of motion) without
fixing the time gauge, i.e. the corresponding lapse function. The key point in
our analysis consists of introducing generic Misner-Chitr\'e-like variables
containing an arbitrary function, whose specification allows one to set up the
same dynamical scheme in any time gauge.
