Heres a video showing how to julienne an onion.

http://www.bing.com/videos/watch/video/how-to-julienne-an-onion/4db8ae676fa4ebba87f04db8ae676fa4ebba87f0-405958689409?q=how+to+julienne+an+onion&FORM=VIRE2

I've heard, and it may be one of those apocryphal "tips" you get from TV food shows that may not be as true as they think....but.....

I have heard that slicing lengthwise gives you finer slices that hold up better, if that's your goal for the final dish. Like for a French Onion soup, where you want to really see the onion strings. Slicing crosswise, the slices will break down more in the cooking process. As Verdigris, says, its related to the amount of damage done to the onion's cellular structure. I usually do crosswise for most applications, but do find I can get consistently thinner slices if I go lengthwise.

The macrobiotics of the '70s taught that onions should be sliced end to end to gain a balance of yin and yang, and I find it easier to handle the onion that way. My husband wants actual rings so he slices crosswise.

I've wondered what a "julienned" onion looks like, given that it is a sphere. ;o)

A julienne?

Slicing lengthwise will produce a milder onion tasting result than slicing crosswise. Slicing crosswise will damage more onion cells there by releasing the chemicals responsible for the pungent odors and flavors. So, I look at the end result I want in the dish and slice accordingly. Also, I don't thing that there is an "industry standard" for the meaning of sliced onions. Some authors detail the meaning of their terms, but that is not a universal standard.

I only slice crosswise if specified

That's what I take it to mean. I only slice crosswise if it specifies to do so.