Note the apostrophe in People’s Republic. You cannot flip the terms, because the relationship between people and republic is defined by that apostrophe. This is a particular kind of republic—a republic which in some sense ‘belongs to’ the people who live there.
Note the quotation marks around “Donetsk People’s Republic”. These mark the phrase within the marks as a distinct structural unit. You cannot move a term outside the quotation marks without changing the sense.
And notice the capital letters. These mark the phrase as a proper noun—a name. You cannot extract a piece of the name and treat it as a common noun.
People’s Republic is a term which was frequently incorporated in the names of states which emerged in the middle years of the 20th century. It designates a state which explicitly claims to govern on behalf of The People as a whole. In some cases these are liberal democracies—the People’s Republic of Bangladesh, for instance—but the word is primarily associated with Marxist states such as the People’s Republic of China.
In the US the term is often used ironically to designate regions dominated by left-wing political parties. A former roommate of mine used to call his home state, affectionately, the People’s Republic of Minnesota.
The “Donetsk People’s Republic” is printed thus, enclosed in quotation marks, because it is not a ‘recognized’ state but the name under which activists in the eastern part of Ukraine declare their secession. The name is undoubtedly chosen to resonate with the domination of Russia in the Soviet era, when many of the Soviet satellites used the term: Hungary, Poland, Bulgaria, and even Ukraine itself for a brief period before it was incorporated in the U.S.S.R.