I have always thought that Aleinu was a piyyut (liturgical poem), not a "prayer" since it does not have a bracha in it. I was surprised to find that the Wikipedia page for piyyut does not list it, ...
It has been pointed out to me that the two halves of the "Alenu" prayer follow a pretty strict pattern. Each contains eight parallel couplets, followed by a Biblical quote. I broke it all down myself, ...