Here's a new style vertical week list
, which is adapted from the week list
style found in the TikZ source code. The day labels are adapted from the answer to Weekday captions with the TikZ library "calendar".

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calendar}
\makeatletter
\tikzstyle{week list}=[%
execute before day scope={%
\ifdate{day of month=1}{\ifdate{equals=\pgfcalendarbeginiso}{}
{%
% On first of month, except when first date in calendar.
\pgfmathsetlength{\pgf@x}{\tikz@lib@cal@month@xshift}%
\pgftransformxshift{\pgf@x}
}%
}{}%
},
execute at begin day scope={%
\pgfmathsetlength\pgf@y{\tikz@lib@cal@yshift}%
\pgf@y=\pgfcalendarcurrentweekday\pgf@y%
\pgftransformyshift{-\pgf@y}%
},
execute after day scope={%
\ifdate{Sunday}{%
\pgfmathsetlength{\pgf@x}{\tikz@lib@cal@xshift}%
\pgftransformxshift{\pgf@x}
}{}%
}
]
\tikzoption{day headings}{\tikzstyle{day heading}=[#1]}
\tikzstyle{day heading}=[]
\tikzstyle{day letter headings}=[
execute before day scope={ \ifdate{day of month=1}{%
\pgfmathsetlength\pgf@ya{\tikz@lib@cal@yshift}%
\pgfmathsetlength\pgf@xa{\tikz@lib@cal@xshift}%
\pgftransformxshift{\pgf@xa}
\foreach \d/\l in {0/M,1/T,2/W,3/T,4/F,5/S,6/S} {
\pgf@ya=\d\pgf@ya%
\pgftransformxshift{-\pgf@xa}%
\pgftransformyshift{-\pgf@ya}%
\node[every day,day heading, anchor=base]{\l};%
}
}{}%
}%
]
\makeatother
\begin{document}
\begin{tikzpicture}
\calendar[
dates=1997-06-01 to 1997-06-last,
week list,
day letter headings,
day heading/.style={gray, xshift=-1em},
month label above left,
month text=\%mt \%y0
];
\end{tikzpicture}
\end{document}