  The Cauchy problem for first order system $L(t, x, \D_t, \D_x)$ is known to
be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \xi)$
which is smooth in $\xi$ and Lipschitz continuous in $(t, x)$. This paper
contains three main results. First we show that a Lipsshitz smoothness globally
in $(t,x, \xi)$ is sufficient. Second, we show that the existence of
symmetrizers with a given smoothness is equivalent to the existence of
\emph{full symmetrizers} having the same smoothness. This notion was first
introduced in \cite{FriLa1}. This is the key point to prove the third result
that the existence of microlocal symmetrizer is preserved if one changes the
direction of time, implying local uniqueness and finite speed of propagation.
