  We present (bi-)symmetric generating functions for the joint distributions of
Euler-Stirling statistics on permutations, including the number of descents
($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of
left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima
($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We
also show how they recover the classical symmetric generating function of
permutations due to Carlitz, Roselle and Scoville (1966).
  Our proofs exploit three different recursive constructions of inversion
sequences, bijections on the multiple equidistributions of Euler-Stirling
statistics over permutations and transformation formulas of basic
hypergeometric series. Furthermore, we establish a new quadruple
equidistribution of Euler-Stirling statistics over inversion sequences, as
progress towards a conjecture proposed by Schlosser and the author (2020).
