Mu-involutions interpolate between permutations and involutions. They index Borel orbits in the variety of complete quadrics, or equivalently certain subvarieties of the flag variety. We study several of their Coxeter theoretic properties: strong and weak Bruhat order, an exchange lemma, and atoms. We then develop a combinatorial recurrence for cohomology representatives of their associated orbits. This is joint work with Zach Hamaker.