I was too lazy to do this with titlesec, but it is probably possible. Instead, I knitted together a tikz-based approach:
\documentclass[twoside]{article}
\usepackage{tikz}
\usetikzlibrary{tikzmark}
\usepackage{tikzpagenodes}
\newcommand{\verticalsection}[2]{%
\stepcounter{section}%
\addcontentsline{toc}{section}{#1 (#2)}%
\tikzmarknode{sec-\thesection-mark}{\strut}%
\checkoddpage%
\ifoddpage%
\begin{tikzpicture}[remember picture, overlay]
\node[anchor=south west, align=left, rotate=-90] at (sec-\thesection-mark.north -| current page marginpar area.west) {\textcolor{purple}{\MakeLowercase{\scshape#1}}\ \MakeLowercase{\scshape#2}};
\end{tikzpicture}%
\else%
\begin{tikzpicture}[remember picture, overlay]
\node[anchor=south east, align=right, rotate=90] at (sec-\thesection-mark.north -| current page marginpar area.east) {\textcolor{purple}{\MakeLowercase{\scshape#1}}\ \MakeLowercase{\scshape#2}};
\end{tikzpicture}%
\fi%
}
\usepackage{lipsum}
\begin{document}
\verticalsection{Lipsuma}{Amuspil foo}
\lipsum[1]
\verticalsection{Lipsumb}{Bmuspil bar}
\lipsum[1]
\newpage
\verticalsection{Lipsumc}{Cmuspil baz}
\lipsum[1]
\end{document}
Odd pages (sections may start at any position on the page):

Even pages:

Another more basic approach using \marginpar:
\documentclass[twoside]{article}
\usepackage{graphicx, xcolor}
\newcommand{\verticalsection}[2]{%
\stepcounter{section}%
\addcontentsline{toc}{section}{#1 (#2)}%
\marginpar[%
\raggedleft\vskip5pt\rotatebox{90}{\parbox{10em}{\raggedleft%
\textcolor{purple}{\MakeLowercase{\scshape#1}}\par\MakeLowercase{\scshape#2}\hskip5pt}%
}%
]{%
\rotatebox{270}{\hskip5pt\parbox{10em}{%
\textcolor{purple}{\MakeLowercase{\scshape#1}}\par\MakeLowercase{\scshape#2}}%
}%
}%
}
\usepackage{lipsum}
\begin{document}
\verticalsection{Lipsuma}{Amuspil foo}
\lipsum[1]
\verticalsection{Lipsumb}{Bmuspil bar}
\lipsum[1]
\newpage
\verticalsection{Lipsumc}{Cmuspil baz}
\lipsum[1]
\end{document}
With pretty much the same output:


You need to take care of the width of the vertical box though: If it is longer than the paragraph it is attached to, the next section title will be shifted downwards.
Also, if a sections starts too far down the page, it may be better to let it start on the next page, since otherwise the section title will protrude over the lower page margin.