Below I defined an alternative equation counter, and use \useoriginaleqn and \usealternateeqn to switch between using either. The switch also updates the representation:

\documentclass{article}
\usepackage{mathtools}
\makeatletter
\newcounter{altequation}
\let\c@equationstore\c@equation
\let\c@altequationstore\c@altequation
% Switch between different equation counters/representations
\newcommand{\useoriginaleqn}{%
\let\c@equation\c@equationstore
\renewcommand{\theequation}{\arabic{equation}}}
\newcommand{\usealternateeqn}{%
\let\c@equation\c@altequation
\renewcommand{\theequation}{\roman{equation}}}
\makeatletter
\AtBeginDocument{\def\theHequation{\theHsection.\theequation}}
\begin{document}
See the following equations: \eqref{eqn:first}, \eqref{eqn:second}, \eqref{eqn:third}, \eqref{eqn:fourth}
\begin{equation} y = ax + b \label{eqn:first} \end{equation}
\usealternateeqn
\begin{equation} E = mc^2 \label{eqn:second} \end{equation}
\begin{equation} S = r^2 \label{eqn:third} \end{equation}
\useoriginaleqn
\begin{equation} y = x^2 \label{eqn:fourth} \end{equation}
\usealternateeqn
\begin{equation} f(x) \vcentcolon= 4x \end{equation}
\useoriginaleqn
\begin{equation} abc \end{equation}
\begin{equation} bca \end{equation}
\begin{equation} cab \end{equation}
\end{document}