Counters and their “partners”

Question

A counter created via \newcounter{<counter>} is stored internally as \c@<counter>, just like a reference made using \label{<label>} is stored internally using \r@<label>. What's the use of \cl@<counter> and \p@<counter>? The macros associated with their use include (from latex.ltx):

\def\@definecounter#1{\expandafter\newcount\csname c@#1\endcsname
     \setcounter{#1}\z@
     \global\expandafter\let\csname cl@#1\endcsname\@empty
     \@addtoreset{#1}{@ckpt}%
     \global\expandafter\let\csname p@#1\endcsname\@empty
     \expandafter
     \gdef\csname the#1\expandafter\endcsname\expandafter
          {\expandafter\@arabic\csname c@#1\endcsname}}
%...
\def\refstepcounter#1{\stepcounter{#1}%
    \protected@edef\@currentlabel
       {\csname p@#1\endcsname\csname the#1\endcsname}%
}
%...
\def\stepcounter#1{%
  \addtocounter{#1}\@ne
  \begingroup
    \let\@elt\@stpelt
    \csname cl@#1\endcsname
  \endgroup}
%...
\def\@addtoreset#1#2{\expandafter\@cons\csname cl@#2\endcsname {{#1}}}

Answer 1

The answer is given by the LaTeX source documentation:

\p@foo Macro that expands to a printed ‘reference prefix’ of counter foo. Any \ref to a value created by counter foo will produce the expansion of \p@foo\thefoo when the \label command is executed. See file ltxref.dtx for an extension of this mechanism.

\cl@foo List of counters to be reset when foo stepped. Has format \@elt{countera}\@elt{counterb}\@elt{counterc}.

For instance,

\newcounter{subsubsection}[subsection]

calls

\@newctr{subsubsection}[subsection]

and this is defined as

\def\@newctr#1[#2]{%
  \@ifundefined{c@#2}{\@nocounterr{#2}}{\@addtoreset{#1}{#2}}}

So the counter subsection is automatically reset when \subsubsection is called, hence \cl@subsubsection contains \@elt{subsection}.

As the documentation says, the p@ feature can be used to "extend" the label of a counter. For instance, article.cls contains

\renewcommand\p@enumii{\theenumi}
\renewcommand\p@enumiii{\theenumi(\theenumii)}
\renewcommand\p@enumiv{\p@enumiii\theenumiii}

with the effect that

\documentclass{article}
\begin{document}
\begin{enumerate}
\item foo 
  \begin{enumerate}
  \item bar
    \begin{enumerate}
    \item\label{itembaz} baz
    \end{enumerate}

  \end{enumerate}

\end{enumerate}

\ref{itembaz}

\end{document}

will result in

and not just "i".