Songs that fool you about where the downbeat is.

Yes! Considered that

The intro to SOS feels like it begins on the 2, but if you're going to count straight fours until the vocal comes in right after the 1, then it starts on the 3! Or you could say that the intro drops a beat before the vocal comes in.

There is a great Switched on Pop episode about Charlie Puth's "Boy" that talks about its "downbeat deception." Listen to the first 10 minutes of this:

https://switchedonpop.com/episodes/87-the-pure-pop-of-charlie-puth-carly-rae-ft-hanif-abdurraqib

Here's the song:

https://www.youtube.com/watch?v=iILJvqrAQ_w

That's a good one, although the backbeat coming in helpfully lines it up.

Just relistened to that Switched on Pop and if you go to minute 16 they also play a bunch of examples of what they dub downbeat deception:

Ismael Miranda – Recordando
The Beatles – She’s a Woman
Sharon Jones & The Dap Kings – Nobody’s Baby
The Cars – Since You’re Gone
Ludwig van Beethoven – Symphony No 5 in C Minor, I

Is there a thread for rap songs with raps on the off-beat?

https://www.youtube.com/watch?v=N_QcvPwa-l0

Jane Weaver - Electric Mountain

https://www.youtube.com/watch?v=_lcFq7Rtx0M

I've always had major trouble hearing where the 1 is on Missy Elliott's "Get Ur Freak On." Intellectually I know it's on the very first kick drum beat, but unless I force myself to count along from that point, I hear the first three counts as "pickup" and hear the 1 as falling on the actual 4 (this is if I'm counting along at the full 178 bpm as opposed to 89 bpm half-time). In other words, I hear the 1 as falling on the 6th note of the iconic tumbi riff.

(which also falls on the third kick drum hit)