  We classify all Hopf algebras which factorize through two Taft algebras
$\mathbb{T}_{n^{2}}(\bar{q})$ and respectively $T_{m^{2}}(q)$. To start with,
all possible matched pairs between the two Taft algebras are described: if
$\bar{q} \neq q^{n-1}$ then the matched pairs are in bijection with the group
of $d$-th roots of unity in $k$, where $d = (m,\,n)$ while if $\bar{q} =
q^{n-1}$ then besides the matched pairs above we obtain an additional family of
matched pairs indexed by $k^{*}$. The corresponding bicrossed products (double
cross product in Majid's terminology) are explicitly described by generators
and relations and classified. As a consequence of our approach, we are able to
compute the number of isomorphism types of these bicrossed products as well as
to describe their automorphism groups.
