Images from this Bulletin illustrate this article. There is no uniformity except that the wings appear up or down. The eagles themselves are as different in graphic design as the language and customs of the countries they represent. Below are listed Supreme Councils with wings up and down from the December 1972 article. Phillipines, Mexico, Paraguay, Belgium, Turkey, France, Iran, Israel, Chile, Brazil, Columbia, Italy, America Southern Jurisdiction, America Northern Masonic Jurisdiction-Honorary Member.
Research reveals no clear knowledge of why wings are up or down.
However, there seems to be some relation of the image to those Supreme Councils who were born from the Masonic roots of Grand Lodges. It appears that for those Supreme Councils which have ties from the Grand Lodge of England, their seals show the wings up, while those which have ties to the Grand Lodge of France, their seals show the wings down. There is a pattern, whether it is intentional or not. Possibly, you saw the double eagle on your father's ring as a child. It did then and does now demand a second and third look.