  Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr
k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ 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 $S_n(t) =
\sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) =
\sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) =
\sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give
bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) -
2nt_n(t^2)$$ 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 $([n], E)$.
