Q:

Looseness for Bezier controls.  Or how to bend in circularly symmetric tikz-drawings (without bend left, etc.) 

If I want to draw a something with circular symmetry, e.g joining pair of  points by arcs as shown in the picture with only two, I face the problem of not being able to use the edge bending.  This is because it implies the choice of bend left (or right, or...) which breaks the symmetry.  
I want to bend my edges to the center of the circle, but the arcs bend too much towards the center (arc in blue).  I can correct, e.g. by adding a coordinate (arc in red).  Isn't there an option to tell controls .. not to bend that much? 
At the bottom there is another picture which shows what I don't want (and happens if I use bend left).  I want in all arcs to be pulled to the center.   I cannot fix by reordering a big list manually in finite time.  

\documentclass{standalone} 
\usepackage{tikz}

\usepackage{ifthen}

\usepackage{pgfplots} 
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}[scale=1]

\coordinate (O) at (0,0);
\draw (O)  circle (4);
%  
 \foreach \i in {1,2,...,8} {

 \node[draw,fill=white,circle] (p\i) at ($(O)+(360*\i/8:4)$)  {\i};
   }

\draw [blue] (p1) .. controls (O) ..  (p3); 
\draw [red] (p1) .. controls  ($(O)+.3*(p2)$)  ..  (p3); 

\end{tikzpicture}

\end{document}

By using, say 
\path [violet, bend left=30, looseness=1.3]
(p1) edge (p2)
(p3) edge (p4)
(p5) edge (p6)
% since I cannot order points, I'll have 
% pairings like the next one:
(p8) edge (p7)
;

This is what I meant with breaking of symmetry: one of the edges 
bends the other side, and not towards the center.  I cannot reorder
data in order to invert the bending-side. 

A:

For your question 'Isn't there an option to tell controls .. not to bend that much?',  I guess the answer is No.  That is because bend left and looseness influence how TikZ creates the control points for you.  If you choose the control points yourself, you bypass that logic.  The path of the curve is then determined by a mathematical formula, so in order for it to look different you have to pick different controls. 
However, you could make the calculation of the intermediate control point a bit easier by using the coordinate system barycentric cs (see also pgfmanual 13.2.2 Barycentric Systems).  Then you don't need to know the coordinate between start and end ((p2) in your example). 
The green lines are all possible connections in your example graph, except loops.  The red/blue lines show how the barycentric coordinate system works when you change the influence of the coordinate (O). 

\documentclass{article} 
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[scale=1]

\coordinate (O) at (0,0);
\draw (O) circle (4);

\foreach \i in {1,2,...,8} {
    \node[draw,fill=white,circle] (p\i) at (360*\i/8:4)  {\i};
}

% Demonstrate influence of O=\looseness in barycentric cs on (p1)--(p3)
\foreach[evaluate={\col={15*\looseness}}] \looseness in {0, 0.25, 0.5, 0.75, 1, 2, ..., 6} {
    \draw [color=blue!\col!red] (p1) .. controls (barycentric cs:p1=0.5,p3=0.5,O=\looseness) ..  (p3); 
}

% Draw all possible edges 
\tikzstyle{every edge}=[
    draw, green,
    to path={
        .. controls (barycentric cs:\tikztostart=0.5,\tikztotarget=0.5,O=0.3) ..
        (\tikztotarget) \tikztonodes
    },
]
\foreach \i in {1, ..., 8}{
    \foreach \j in {1, ..., 8}{
        \ifnum\i=\j\else
            \draw [green] (p\i) edge (p\j); 
        \fi
    }
}

\end{tikzpicture}
\end{document}

A:

With bend left, you should be able to get what you want (unless I totally misundertsood your demand)... but you have to pay attention to the order of the coordinates. 
\documentclass{standalone} 
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}[scale=1]

\coordinate (O) at (0,0);
\draw (O)  circle (4);
%  
 \foreach \i in {1,2,...,8} {

 \node[draw,fill=white,circle] (p\i) at ($(O)+(360*\i/8:4)$)  {\i};
   }
\foreach \i/\j in {1/3,2/4,3/5,4/6,5/7,6/8,7/1,8/2}{
\draw [blue] (p\i) .. controls (O) ..  (p\j); 
\draw [red] (p\i) .. controls  ($(O)+.3*(p2)$)  ..  (p\j); 
\draw [green,bend left,looseness=1.5] (p\i) edge (p\j); 
}
\end{tikzpicture}

\end{document}

EDIT A proposal with number sorting so that the actual plot is always done in increasing node numbers. 
\documentclass{standalone} 
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}[scale=1]

\coordinate (O) at (0,0);
\draw (O)  circle (4);
%  
 \foreach \i in {1,2,...,8} {

 \node[draw,fill=white,circle] (p\i) at ($(O)+(360*\i/8:4)$)  {\i};
   }
\foreach \ii/\jj [
    evaluate={\ii as \i using ifthenelse(\jj<\ii,int(\jj),int(\ii))},
    evaluate={\jj as \j using ifthenelse(\jj<\ii,int(\ii),int(\jj))},
] in {1/2,3/4,5/6,7/8,7/6}{
\draw [blue] (p\i) .. controls (O) ..  (p\j); 
\draw [red] (p\i) .. controls  ($(O)+.3*(p2)$)  ..  (p\j); 
\draw [green,bend left,looseness=1] (p\i) edge (p\j); 
}
\end{tikzpicture}

\end{document}

