  As shown by Bousquet-M\'elou--Claesson--Dukes--Kitaev (2010), ascent
sequences can be used to encode $({\bf2+2})$-free posets. It is known that
ascent sequences are enumerated by the Fishburn numbers, which appear as the
coefficients of the formal power series $$\sum_{m=1}^{\infty}\prod_{i=1}^m
(1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose
ascent sequences, which leads to: (i) a calculation of the Euler--Stirling
distribution on ascent sequences, including the numbers of ascents ($\asc$),
repeated entries $(\rep)$, zeros ($\zero$) and maximal entries ($\max$). In
particular, this confirms and extends Dukes and Parviainen's conjecture on the
equidistribution of $\zero$ and $\max$. (ii) a far-reaching generalization of
the generating function formula for $(\asc,\zero)$ due to Jel\'inek. This is
accomplished via a bijective proof of the quadruple equidistribution of
$(\asc,\rep,\zero,\max)$ and $(\rep,\asc,\rmin,\zero)$, where $\rmin$ denotes
the right-to-left minima statistic of ascent sequences. (iii) an extension of a
conjecture posed by Levande, which asserts that the pair $(\asc,\zero)$ on
ascent sequences has the same distribution as the pair $(\rep,\max)$ on
$({\bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition
of $({\bf2-1})$-avoiding inversion sequences parallel to that of ascent
sequences.
  This work is motivated by a double Eulerian equidistribution of Foata (1977)
and a tempting bi-symmetry conjecture, which asserts that the quadruples
$(\asc,\rep,\zero,\max)$ and $(\rep,\asc,\max,\zero)$ are equidistributed on
ascent sequences.
