  We reconsider the gauge symmetries of the spinning particle by a direct
examination of the Lagrangian using a systematic procedure based on the Noether
identities. It proves possible to find a set of local Bosonic and Fermionic
gauge transformations that have a simple gauge group structure, which is a true
Lie algebra, both for the massless and massive case. This new Fermionic gauge
transformation of the "position" and "spin" variables in the action decouples
from that of the "einbein" and "gravitino".
  It is also possible to redefine the fields so that this simple algebra of
commutators of the gauge transformations can be derived directly starting from
the Lagrangian written in these new variables.
  We discuss a possible extension of our analysis of this simple model to more
complicated cases.
