Reverse TikZ stroke direction to use with fill rules

Question

This question is inspired by my own answer: How to draw the crossings in a tikz picture? where I draw the blue ellipse in four stages. I was thinking that it should be possible to draw with only one \fill command. To make the correct \clip, I need to understand the default \clip or \fill non-Zero Winding Rule, the nonzero rule (the manual 15.5.2 page 181) and not the even odd rule.

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In this example, we can see that circles(and ellipses) are drawn with reverse stroke compared to rectangles:

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
 \draw[fill=red]  (0,0) circle (1) (0,0) rectangle (2,2);
 \draw[fill=green] (4,-1) rectangle (6,1) (5,0) rectangle (7,2);
 \draw[fill=blue] (9,0) circle (1) (9.5,0.5) circle (1);
\end{tikzpicture}
\end{document}

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To make the correct clip, I need to select an area from four overlapping ellipses and a rectangle. I believe that it is not possible without altering the stroke direction of some of the ellipses!? It would not help to make clip use even odd rule.

Question: How do I not fill the intersection of two overlapping circles(ellipses) without the even odd rule?

MWE:

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
 \fill (0,0) circle (2) (1,1) circle (2);
\end{tikzpicture}
\end{document}

Desired output:

I can not find anything in the manual about changing the stroke direction. Can it be done? Maybe there is an other way?

Edit: I just found that for my original purpose, it does help to make clip use even odd rule (can be done with a scope). -but my question is still the same -is it possible without.

Answer 1

My spath3 library was designed to do low level manipulation of paths, including reversing the stroke direction. The public interface is a little basic as the intention was to provide functionality that could be used in other packages (such as the knot and calligraphy TikZ libraries). Here's a proof of concept.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/568999/86}
\usepackage{tikz}
\usepackage{spath3}
\begin{document}
\begin{tikzpicture}
\path[save spath=circle1]  (0,0) circle[radius=2];
\path[save spath=circle2] (1,1) circle[radius=2];
 \fill (0,0) [insert spath=circle1] (1,1) [insert reverse spath=circle2];
\end{tikzpicture}
\end{document}

Edit by OP: I believe that spath3 syntax has been updated. A working code is now:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{spath3}
\begin{document}
\begin{tikzpicture}
\path[spath/save=circle1]  (0,0) circle[radius=2];
\path[spath/save=circle2] (1,1) circle[radius=2];
\fill (0,0) [spath/insert=circle1] (1,1) [spath/insert reverse=circle2];
\end{tikzpicture}
\end{document}

Answer 2

Stroke direction of rectangle is relative to the provided points. For example, in your first code, try (5,2) rectangle (7,0) instead of (5,0) rectangle (7,2). With circle, you can use circle(-2 and 2) instead of circle(2).

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
  \draw[fill=red]  (0,0) circle (1) (0,0) rectangle (2,2);
  \draw[fill=green] (4,-1) rectangle (6,1) (5,2) rectangle (7,0);
  \draw[fill=blue] (9,0) circle (1) (9.5,0.5) circle (-1 and 1);
\end{tikzpicture}
\end{document}