I'm gonna disagree with IP here, even though he's a Nyawker and I am not....
The name of the street is East 20th street or West 20th Street. The direction is a necessary part of the name. Here's why:
If you are going north on 5th Ave and want to go to 25 East 20th Street (
http://snipurl.com/l4v3 ), you would turn right onto East 20th Street; but there's also a 25 West 20th Street (
http://snipurl.com/l4uz ) and you would turn left from Fifth Avenue to get there.
Both those addresses have the same street number, so you cannot disregard the East and West designations.
I'm curious, though, as to why they decided to put east and west on opposite sides of 5th Avenue, rather than First Avenue. In Denver, East and West are divided by Broadway, while North and South are divided by Ellsworth. So if yu wanted to go to say the 700 Block of East Sixth Street you would go seven blocks East of Broadway and six blocks north of Ellsworth.
The situation becomes much more complicated elsewhere. Look at this map of Salt Lake City, UT
http://snipurl.com/l4vhNote that there is a W North Temple, and E North Temple, a W South Temple, and though not shown on this map, probably an E South Temple. There is also A S West Temple and an N West Temple.
But beyond that, look at W 400 N and W 400 S. They run E-W parallel to one another but 8 blocks apart. And W 400 N crosses N 400 W and W 400 S crosses both S 400 W as well as S 400 E.
And then there would be house numbers in front of the name of the street.
I wonder if the Mormons did this so anyone invading SLC would be so hopelessly lost that they would have to ask for directions, giving the defenders a great chance to take them out.