We sing the Asoyin - Agomado and Sarago songs starting on the first double of the bell, or the 3 pulse.
Interesting that you're saying it's on another beat.
One of my favorite mind benders comes from the CD Asyoyi by Lazaro Ros - trying to sing Ferekete Mayinoma Mayinokan Ferekete.... So that it starts on the four pulse. Ouch! It hides the one so well, that if you're playing bell, the four becomes your one.. Totally deadly.. I love it!
So, in support of the question in the original post, that's one of the most intriguing reasons for my own pursuit in attempting to learn this music.
The rhythmic tricks played by all of the parts, including the song and how they hide or diversify the beat. It's so wonderful,, Each song is like a diamond!