Wakizashi is typically worn beside/over the katana. At least in every drawing/painting I've ever seen from classical Japan. Both go edge up ("blade up" is physically impossible, if the sword is to stay in the scabbard), because it is easier to draw the sword in a fluid manner thanks to the curve--to draw and cut in one motion--as the scabbard is basically tucked into the belt, rather than hanging from the belt in the Western style.
Oh, and for note, the katana blade is roughly the same length as that of a broadsword. The balance is different, though because the hilt is more than twice as long. The same goes for the wakisashi v. Western "shortsword".
Miyamoto Musashi states that "the abdomen is braced by the scabbard of the short sword in such a manner that the belt does not loosen" (The Book of Five Rings, "The Water Scroll", 19, trans.Thomas Cleary).
Longsword v. broadsword is also a sort of interesting debate, since they didn't really exist concurrently. However, most fantasy uses the terms interchangably because Tolkien did, whereas the fencing manuals of Europe make a clear division--the longsword having a narrower blade and a point, while the broadsword being a heavier blade with a rounded tip that seems to have been phased out of use by the 16th century. After about the 15th to 16th centuries, the manuals don't even mention broadswords, to the best of my knowledge.