I just learned the phrase "que no se te olvide ..." from that new Selena Gomez song. It seems to be an emphatic "don't you forget it". It took me awhile to wrap my head around the grammar -- the subject is the thing you shouldn't forget, the verb is olvidarse in subjunctive, and the indirect object is te.
This seems like the complete reverse of the phrase "no te olvides el paraguas" where the subject is tú and the direct object is el paraguas. Are there many other verbs in Spanish where the subject and object can be swapped like this and keep the same basic meaning?
I'm familiar with impersonal se and passive se, but this seems like something different, is there another name for this type of phrase?
Also, I can't seem to find any examples of a sentence beginning with "No se te olvide", why do sentences always start with "Que no se te olvide" or "Y no se te olvide", is it just because the subjunctive case necessitates a conjunction?