  We give a complete classification of conformally covariant differential
operators between the spaces of differential $i$-forms on the sphere $S^n$ and
$j$-forms on the totally geodesic hypersphere $S^{n-1}$ by analyzing the
restriction of principal series representations of the Lie group $O(n+1,1)$.
  Further, we provide explicit formul\ae{} for these matrix-valued operators in
the flat coordinates and find factorization identities for them.
