Kari:
As Leonid has rightly pointed out, today's Asscher means a very different thing from the original (now called "antique Asscher" ) from the early part of this century. If I have my information right, here's the deal (and mind you, I'm no expert, just reporting what I've come across in my own research):
The original Asscher is a stepped square cut with a high crown and small table. What that translates into, with those cut corners, is what almost looks like a stop sign on top, even if it keeps a relatively square shape at the girdle. These are pretty hard to find, especially in very white stones with good clarity. They can be very costly and are prized by antique diamond collectors. Today's Asscher, also called the "modern Asscher" at least by the people I spoke with, is really just a square emerald cut. They aren't as expensive. The Royal Asscher, as Leo pointed out, is a branded square cut with more fire due to more pavilion facets. It's only available through licensed retailers and it's pricey.
For more info, go here:
http://www.chubbcollectors.com/Vacnews/index.jsp?form=2&ArticleId=25
Hope this helps!
--TheRock