Bijective proofs for Eulerian numbers in types B and D

Let $\left\langle{n\atop k}\right\rangle$, $\left\langle{B_n\atop k}\right\rangle$, and $\left\langle{D_n\atop k}\right\rangle$ be the Eulerian numbers in the types $A$, $B$, and $D$, respectively -- that is, the number of permutations of $n$ elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type $B$ descents, the number of even signed permutations (of $n$ elements) with $k$ type $D$ descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \left\langle{n\atop k}\right\rangle t^k$, $B_n(t) = \sum_{k = 0}^{n}\left\langle{B_n\atop k}\right\rangle t^k$, and $D_n(t) = \sum_{k = 0}^{n}\left\langle{D_n\atop k}\right\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2ntS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t)\ .$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.