  The Birkhoff polytope is defined to be the convex hull of permutation
matrices, $P_{\sigma}\ \forall \sigma\in S_n$. We define a second-order
permutation matrix $P^{[2]}_{\sigma}$ in $\mathbb{R}^{n^2\times n^2}$
corresponding to a permutation $\sigma$ as $(P^{[2]}_{\sigma})_{ij,kl} =
(P_{\sigma})_{ij}(P_{\sigma})_{kl}$. We call the convex hull of the
second-order permutation matrices, the {\em second-order Birkhoff polytope} and
denote it by ${\cal B}^{[2]}$. It can be seen that ${\cal B}^{[2]}$ is
isomorphic to the QAP-polytope, the domain of optimization in {\em quadratic
assignment problem}. In this work we revisit the polyhedral combinatorics of
the QAP-polytope viewing it as ${\cal B}^{[2]}$. Our main contribution is the
identification of an exponentially large set of new facets of this polytope.
Also we present a general inequality of which all the known facets of this
polytope as well as the new ones, that we present in this paper, are special
instances. We also establish the existence of more facets which are yet to be
identified.
