Palindromic Poincaré polynomials in Type $B$

We completely describe the set of all palindromic Poincaré polynomials in type $B_n$. Additionally, we show that all lower Bruhat intervals in type $A_n$ occur in type $B_n$. As a consequence, all Poincaré polynomials $P_w(q)$ and all Kazhdan–Lusztig polynomials $P_{x,y}(q)$ of type $A_n$ occur in type $B_n$.