While we're at it, I hope you'll answer my other question: New Yorkers refer to numbered street intersections, such as "3rd and 21st," without specifying which are Avenues and which are Streets. I'm assuming Avenues run in one direction and Streets in the other. The question is: what convention are they all using, which is unstated but universally understood? My assumption is that it's "Avenues, then Streets," because that's alphabetical.
Yes, when they use that shorthand, they are referring to streets in Manhattan, and if they’re First through 10th they are avenues. The avenues, plus Broadway and a couple of smaller fringe streets, run north to south. The streets go up to something like 244th, including part of the Bronx. They generally use the street (or avenue) that the address is actually on before giving the cross street. And you’re right, if simply referencing an intersection, the avenue comes first. They’ll slap an “Ave.” and/or a “St.” on if it’s open to confusion.