  Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and
suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the
Yetter--Drinfel'd category ${}_H^H{\mathcal YD}$. We describe the bialgebras
$(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$
and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in
${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in
${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twist
bialgebra structures on the tensor product of bi-products.
