  We define symmetric Dellac configurations as the Dellac configurations that
are symmetric with respect to their centers. The symmetric Dellac
configurations whose lengths are even were previously introduced by Fang and
Fourier under the name of symplectic Dellac configurations, to parametrize the
torus fixed points of symplectic degenerate flag varieties. In general,
symmetric Dellac configurations generate the Poincar\'e polynomials of (odd or
even) symplectic or orthogonal versions of the degenerate flag varieties. In
this paper, we give several combinatorial interpretations of the polynomial
extensions $(D_n(x))_{n \geq~0}$ of median Euler numbers, defined by
Randrianarivony and Zeng, in terms of objects that we name extended Dellac
configurations and which generate symmetric Dellac configurations. As a
consequence, the cardinalities of the odd and even symmetric Dellac
configurations are respectively given by the two adjoining sequences $(l_n)_{n
\geq~0} = (1, 1, 3, 21, 267,\dots)$ and $(r_n)_{n \geq~0} =
(1,2,10,98,1594,\dots)$, defined as specializations of the polynomials
$(D_n(x))_{n \geq~0}$.
