Surprise 1 - the solution

But? ... the refrain of this song, called "When I Hear a Syncopated Tune" is not syncopated! Yes, there is a syncopation in the final turn of the refrain, which is repeated three times—but the "earworm" melody of the refrain does not want to follow its lyrics. For whichever reason.

By the way, it is hard to believe that Mr. Hirsch (who certainly knew all about syncopations—we are at the end of ragtime's high time!) wanted to present an intricate contradictory riddle to his audience. With this light entertaining piece, I'd rather assume that the text was written after the melody had been composed ... or something like that. But I do not know the historical facts.

The refrain of this song can also be heard in this video posted at youtube , starting around 3:33.

Does also this song, called "Those Ragtime Melodies" subvert its title? I'm not so sure here: Nowadays, a "ragtime" must be syncopated in Scott-Joplin-ish manner—but it might well have been that many songs around 1900 were sold under this heading even though they were not syncopated. Still, also this song surprised me.