Touching arrows in middle point

Question

I try to draw something with six nodes and four arrows. The arrows should touch in the most bended point - in the middle of the arrows.

However, I don't want to play around with moving nodes so that the arrows touch by coincidence. What is the best way?

So far I have this:

\documentclass[tikz,border=3mm]{standalone}
 \usetikzlibrary{positioning}
 \begin{document}
 \begin{tikzpicture}
 \node[](A){A};
 \node[below = of A](B){B};
\node[left = of A](C){C};
\node[right = of A](D){D};
\node[left = of B](E){E};
\node[right = of B](F){F};
\draw [->] (B.east) to [bend right] (A.east);
\draw [->] (F.west) to [bend left] (D.west);
\draw [->] (A.west) to [bend right] (B.west);
\draw [->] (C.east) to [bend left] (E.east);

\end{tikzpicture}
\end{document}

Answer 1

With barycentric coordinates you can determine a good candidate for the touching point, which can be used in various constructions such as

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{positioning}
\begin{document}
\begin{tikzpicture}
\begin{scope}
 \node(A){A};
 \node[below = of A](B){B};
 \node[left = of A](C){C};
 \node[right = of A](D){D};
 \node[left = of B](E){E};
 \node[right = of B](F){F};
 \path (barycentric cs:A=1,B=1,D=1,F=1) coordinate (ABDF)
     (barycentric cs:A=1,B=1,C=1,E=1) coordinate (ABCE);
 \draw[->] (B.east) to[out=45,in=-90] (ABDF) to[out=90,in=-45] (A.east);
 \draw[->] (F.west) to[out=135,in=-90] (ABDF) to[out=90,in=-135] (D.west);
 \draw[->] (A.west) to[out=-135,in=90] (ABCE) to[out=-90,in=135] (B.west);
 \draw[->] (C.east) to[out=-45,in=90] (ABCE) to[out=-90,in=45]  (E.east);
\end{scope}
\begin{scope}[xshift=5cm]
 \node(A){A};
 \node[below = of A](B){B};
 \node[left = of A](C){C};
 \node[right = of A](D){D};
 \node[left = of B](E){E};
 \node[right = of B](F){F};
 \path (barycentric cs:A=1,B=1,D=1,F=1) coordinate (ABDF)
     (barycentric cs:A=1,B=1,C=1,E=1) coordinate (ABCE);
 \draw[->] plot[smooth] coordinates {(B.east) (ABDF) (A.east)};
 \draw[->] plot[smooth] coordinates {(F.west) (ABDF) (D.west)};
 \draw[->] plot[smooth] coordinates {(A.west) (ABCE) (B.west)};
 \draw[->] plot[smooth] coordinates {(C.east) (ABCE) (E.east)};
\end{scope}
\end{tikzpicture}
\end{document}

Of course one can make the arrow heads more funky etc.