These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Genius Bonus: Things make more sense with knowledge of Mayan mythology and concepts.
Hilarious in Hindsight: Ayato's claim in the first episode that he's "not interested in older women" is much more amusing on the second viewing.
There's much, much more, actually. This anime thrives on making small conversations much more interesting with a second viewing. For example, when Ayato first visits Itsuki's place, Itsuki mentions that he should be like visiting a relatives house. Ayato replies that he's never done that. The irony in this is that unknown to Ayato Itsuki is Ayato's clone and Quon is their biological mother.
The intro sequence to the show is done with the Overture to Die Meistersinger as the background music.
A recurring sequence (arguably a form of leitmotif) is, when Ayato meets Reika, she is in a higher place than he is with her back turned to him, wearing a yellow dress. She than turns towards him and smiles. This is sometimes coupled with a picture of wings behind her that makes it look like she has wings.
She's also wearing a scarf. When caught by the wind, it looks like wings.