Reverse TikZ stroke direction to use with fill rules

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.

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}

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}