  The absolute order on the hyperoctahedral group $B_n$ is investigated. It is
proved that the order ideal of this poset generated by the Coxeter elements is
homotopy Cohen-Macaulay and the M\"obius number of this ideal is computed.
Moreover, it is shown that every closed interval in the absolute order on $B_n$
is shellable and an example of a non-Cohen-Macaulay interval in the absolute
order on $D_4$ is given. Finally, the closed intervals in the absolute order on
$B_n$ and $D_n$ which are lattices are characterized and some of their
important enumerative invariants are computed.
