Let's say I have a syndicated post on site A that has a rel="canonical" towards the original post from site B.
Do the backlinks to this syndicated post bring any value for the site A Domain Authority?
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Nope. A rel="canonical" isn't a backlink.
A rel="canonical" points to the preferred version of a page, its intention is to help search engines with duplicated content. It does not act as a link, but as a "suggestive" redirect.
If a search engine finds that both the canonical page has very similar content to the page on site A, it will only index the canonical page and ignore the page on site A. But if they appear to be quite different, search engines will index both pages.
If both pages do get indexed, a search engine could use the canonical link as a tie breaker when determining which page is more relevant. But this is not PageRank or Domain Authority stuff, this is just dealing with duplicates and ranking the more relevant pages higher.
Please note that canonicals are treated like 301 redirects so it will pass the PageRank to the specified canonical URL.
NO, it only avoids duplicate content for Google. Either it is canonical or not, it do not affect ranking, the links with do follow and nofollow only matters.
No, but they do the other way around. Any links that site A gets on the article then get passed through to site B.