The Bruhat order encodes algebraic and topological information of Schubert varieties in the flag manifold and possesses rich combinatorial properties. In this talk, we discuss some interrelated stories regarding the Bruhat order: self-dual Bruhat intervals, minimal exponents in the Kazhdan-Lusztig polynomials, Billey-Postnikov decompositions and automorphisms of the Bruhat graph. This is joint work with Christian Gaetz.