Eel is a common freshwater fish in Japan and Japanese like eel cuisine. The most common way to eat eel is to grill. Grilled eel on rice is one of the luxurious dishes in Japan which prices are went up in decades due to lack of juvenile fish.
Then, there are two ways to eat grilled eels. The first one is kabayaki style which is dipped into soy sauce based sweeten sauce and grilled. The second one is called shirayaki(white grill), which is not dipped into the sauce. Kabayaki style is much popular than shirayaki and mostly accompanied with rice.
Although grilled eels are eaten nationwide, the way to cook is different in Osaka and in Tokyo. Tokyo style is called Kanto style, due to Tokyo is located at Kanto region and eels are slit down its back and butterflied. Then, the eel is grilled first as shirayaki, without sauce. Later the eel is steamed and finally grilled as kabayaki or shirayaki.
On the other hand, eels are slit down its belly and butterflied. The eel is grilled as shirayaki or kabayaki styles from the beginning and it will not be steamed at all. This method is called Kansai style. Kansai is the region where Osaka and Kyoto are located. Most known reason to avoid slitting down the belly is that slitting down the belly looks similar to harakiri. Tokyo, is successor of Edo, which was ruled by shogun = samurai. However, Osaka and Kyoto were cities of merchants and aristocracy.