What are the LaTeX codes for clockwise and counter-clockwise integrals (∱ and ⨑)?

With the MnSymbol package, you could use the following symbols:

\documentclass{article}

\usepackage{MnSymbol}

\begin{document}

\[
\rcirclerightint
\lcirclerightint
\rcircleleftint
\lcircleleftint
\]

\end{document}

(other package might know these symbols as \ointclockwise and \ointctrclockwise)

---

If you only want half a circle, you can use the mathdesign package:

\documentclass{article}

\usepackage[charter]{mathdesign}

\begin{document}

\[
\intclockwise
\]

\end{document}

\documentclass{standalone}
\usepackage{array,esint}
\def\CMD#1{%
   $ \csname#1\endcsname \displaystyle\csname#1\endcsname $ & \texttt{\textbackslash#1} &}

\begin{document}
\def\arraystretch{2}
\begin{tabular}{@{}*{2}{r@{\kern3pt}l}r@{\kern3pt}l@{}l@{}}
\CMD{int}   \CMD{iint}     \CMD{iiintop} \\
\CMD{iiiintop}\CMD{dotsintop}\CMD{ointop}  \\
\CMD{oiint}   \CMD{sqint}    \CMD{sqiint} \\
\CMD{ointctrclockwise} \CMD{ointclockwise} \CMD{varointclockwise} \\
\CMD{varointctrclockwise} \CMD{fint} \CMD{varoiint}\\
\CMD{landupint} \CMD{landdownint}
\end{tabular}
\end{document}

Although some of the symbols in esint come very close, I don't believe there is any package (for pdfLaTeX) that provides a version of this symbol compatible with the computer modern maths font.

The following is a (very simple) adaption of this answer by Heiko Oberdiek. It can be used to superimpose any symbol on top of an integral sign.

\documentclass{article}

\usepackage{graphicx} %% <- for \resizebox and \rotatebox
\usepackage{amsmath}
\usepackage{amssymb} %% <- for \curverightarrow, \curveleftarrow

\makeatletter %% <- make @ usable in macro names

\let\DOTSI\relax % amsmath support for \dots
\newcommand*{\letteronint}[1]{%
  \DOTSI
  \mathop{%
    \mathpalette\@LetterOnInt{#1}%
  }%
  \mkern-\thinmuskip % thin space is inserted between two \mathop
  \int
}
\newcommand*{\@LetterOnInt}[2]{%
  \sbox0{$#1\int\m@th$}%
  \sbox2{$%
    \ifx#1\displaystyle
      \textstyle
    \else
      \scriptscriptstyle
    \fi
    #2%
  \m@th$}%
  \dimen@=.4\dimexpr\ht0+\dp0\relax
  \ifdim\dimexpr\ht2+\dp2\relax>\dimen@
    \sbox2{\resizebox*{!}{\dimen@}{\unhcopy2}}%
  \fi
  \dimen@=\wd0 %
  \ifdim\wd2>\dimen@
    \dimen@=\wd2 %
  \fi
  \rlap{\hbox to \dimen@{\hfil
    $#1\vcenter{\copy2}\m@th$%
  \hfil}}%
  \ifdim\dimen@>\wd0 %
    \kern.5\dimexpr\dimen@-\wd0\relax
  \fi
}

%% Define arrow curving downwards:
\newcommand*{\curvearrowbotright}{\mathpalette\rotmath@internal\curvearrowleft}
\newcommand*\rotmath@internal[2]{\rotatebox{180}{$\m@th#1#2$}}

\makeatother %% <- revert @

\newcommand*{\intcw}{\letteronint{\mspace{3mu}\curvearrowright}}
\newcommand*{\intccw}{\letteronint{\mspace{3mu}\curvearrowbotright}}


\begin{document}

\[
  \displaystyle      \intcw_a^b f \dots \intccw_a^b f \qquad
  \textstyle         \intcw_a^b f \dots \intccw_a^b f \qquad
  \scriptstyle       \intcw_a^b f \dots \intccw_a^b f \qquad
  \scriptscriptstyle \intcw_a^b f \dots \intccw_a^b f
\]

\end{document}

The \scriptscriptstyle version does not look good, but you probably won't want to use this symbol at that scale.