How to define a symbol similar to Yinyang?

Here is one way to do it using TikZ. Code for the symbol was obtained from this answer, and modified to match the size of \bullet. The size of the symbol is also scalable depending on your font size.

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{tikz}

\newcommand{\customyinyang}[1][1]{%
    \begin{tikzpicture}[scale=#1*0.07]
      \draw[line width = #1*0.05mm,transform canvas={yshift=0.02cm}] (0,0) circle (1cm);
      \path[fill=black,transform canvas={yshift=0.02cm}] (90:1cm) arc (90:-90:0.5cm)
                        (0,0)    arc (90:270:0.5cm)
                        (0,-1cm) arc (-90:-270:1cm);

    \end{tikzpicture}}

\begin{document}

Text \customyinyang[1] $\bullet$

\LARGE Large text \customyinyang[1.8] $\bullet$

\end{document}

UPDATE

A version which scales automatically based on font size. As requested in the comments.

\documentclass{article}
\usepackage{tikz}

\newcommand{\customyinyang}{%
    \begin{tikzpicture}[scale=0.45]
      \draw[line width = 0.05ex,transform canvas={yshift=0.12ex}] (0,0) circle (1ex);
      \path[fill=black,transform canvas={yshift=0.12ex}] (90:1ex) arc (90:-90:0.5ex)
                        (0,0)    arc (90:270:0.5ex)
                        (0,-1ex) arc (-90:-270:1ex);
    \end{tikzpicture}}

\begin{document}

Text \customyinyang~$\bullet$

\Huge Huge text \customyinyang~$\bullet$

\end{document}

As asked in comment here a version based on Myles answer which scales the size of the symbol:

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{tikz}
\usepackage{scalerel}

\newcommand{\customyinyang}{%
    \scalerel*{%
        \tikz[anchor=base, baseline]{%
            \draw[line width = 0.01em] (0,0.26em) circle (0.2em);
            \path[fill=black] (90:0.46em) arc (90:-90:0.1em)
                              (0,0.46em)    arc (90:270:0.2em)
                              (0,0.06em) arc (-90:-270:0.1em);
        }%
    }{\bullet}%
}

\begin{document}


{\Huge $F  \customyinyang \bullet$}   

{\normalsize $F \customyinyang \bullet$}   

{\tiny $ F \customyinyang \bullet$}   


\end{document}

You can increase o decrease the size of the symbol with the command \Yiyang with the coordinates of the points and of the circles. After with circle (.25cm) you can create a big, small o little bullet changed the value of (.25cm).

\documentclass[a4paper,12pt]{article}
\usepackage{tikz}
\newcommand{\Yiyang}{\tikz{\begin{scope}
\clip (0,0) circle (.25cm);
\fill[black] (0cm,.25cm) rectangle (-.25cm, -.25cm);
\end{scope}
\fill[black] (0,0.125) circle (0.125cm);
\fill[white] (0,-0.125) circle (0.125cm);
\draw (0,0) circle (.25cm);}
}
\newcommand{\newbullet}{\tikz\draw[black,fill=black] (0,0) circle (.25cm);
}

\begin{document}
Text \Yiyang and text! \ldots and \newbullet

\end{document}