  Some variations of $\pi$-regular and nil clean rings were recently introduced
in \cite{5,8,7}, respectively. In this paper, we examine the structure and
relationships between these classes of rings. Specifically, we prove that $(m,
n)$-regularly nil clean rings are left-right symmetric and also show that the
inclusions ($D$-regularly nil clean) $\subseteq$ (regularly nil clean)
$\subseteq$ ($(m,n)$-regularly nil clean) hold, as well as we answer Questions
1, 2 and 3 posed in \cite{8}. Moreover, some other analogous questions
concerning the symmetric properties of certain classes of rings are treated as
well by proving that centrally Utumi rings are always strongly $\pi$-regular.
