  We study the lifting of the Schubert stratification of the homogeneous space
of complete real flags of $R^{n+1}$ to its universal covering group
$Spin_{n+1}$. We call the lifted strata the Bruhat cells of $Spin_{n+1}$, in
keeping with the homonymous classical decomposition of reductive algebraic
groups. We present explicit parameterizations for these Bruhat cells in terms
of minimal-length expressions $\sigma=a_{i_1}... a_{i_k}$ for permutations
$\sigma\in S_{n+1}$ in terms of the $n$ generators $a_i=(i,i+1)$. These
parameterizations are compatible with the Bruhat orders in the Coxeter-Weyl
group $S_{n+1}$. This stratification is an important tool in the study of
locally convex curves; we present a few such applications.
