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John Lindsay ran as a Republican AND a Liberal, his first time. Koch ran on the Dem and GOP ticket at the same as well, his second time. He was a liberal in name only, basically a moderate Republican, Rudy WAS a Republican. Bloomberg simply switched parties before the election.
The only two mayors who were Dem machine loyalists were Abe Beame and Dave Dinkins, the #1 and #2 worst mayors NYC had since maybe the 19th Century.