  We develop several combinatorial models that are useful in the study of the
$SL_n$-variety $\mathcal{X}$ of complete quadrics. Barred permutations
parameterize the fixed points of the action of a maximal torus $T$ of $SL_n$,
while $\mu$-involutions parameterize the orbits of a Borel subgroup of $SL_n$.
Using these combinatorial objects, we characterize the $T$-stable curves and
surfaces on $\mathcal{X}$, compute the $T$-equivariant $K$-theory of
$\mathcal{X}$, and describe a Bia{\l}ynicki-Birula cell decomposition for
$\mathcal{X}$. Furthermore, we give a computational characterization of the
Bruhat order on Borel orbits in $\mathcal{X}$.
