  Let $1\leq p<\infty$ and let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$
be a bounded map between noncommutative $L^p$-spaces. If $T$ is bijective and
separating (i.e., for any $x,y\in L^p({\mathcal M})$ such that $x^*y=xy^*=0$,
we have $T(x)^*T(y)=T(x)T(y)^*=0$), we prove the existence of decompositions
${\mathcal M}={\mathcal M}_1\mathop{\oplus}\limits^\infty{\mathcal M}_2$,
${\mathcal N}={\mathcal N}_1 \mathop{\oplus}\limits^\infty{\mathcal N}_2$ and
maps $T_1\colon L^p({\mathcal M}_1)\to L^p({\mathcal N}_1)$, $T_2\colon
L^p({\mathcal M}_2)\to L^p({\mathcal N}_2)$, such that $T=T_1+T_2$, $T_1$ has a
direct Yeadon type factorisation and $T_2$ has an anti-direct Yeadon type
factorisation. We further show that $T^{-1}$ is separating in this case. Next
we prove that for any $1\leq p<\infty$ (resp. any $1\leq p\not=2<\infty$), a
surjective separating map $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is
$S^1$-bounded (resp. completely bounded) if and only if there exists a
decomposition ${\mathcal M}={\mathcal M}_1
\mathop{\oplus}\limits^\infty{\mathcal M}_2$ such that $T|_{L^p({\tiny
{\mathcal M}_1})}$ has a direct Yeadon type factorisation and ${\mathcal M}_2$
is subhomogeneous.
