Pigeonholing Hurricanes

I was wondering about this:

If a hurricane, such as Hurricane Dean, starts in the Atlantic Ocean, it hits land in Mexico and may slow down but crosses into the Pacific Ocean and regains hurricane strength, would it keep the name given in the Atlantic Ocean or would it be given the next name on the list on the Pacific side?

This is how the National Hurricane Center has been handling this matter since 2001:

If a hurricane passes from the Atlantic to the Pacific and keeps its basic hurricane shape and circulation pattern and so on, it keeps its Atlantic name.

If the storm loses its hurricane circulation pattern while crossing Mexico or Central America and re-forms in the Pacific either as a tropical storm or a hurricane, it gets the next name on the Pacific list. I think that would be Gil.

Before 2001, they just renamed the storm no matter what.

For example, in 1996 Hurricane Cesar went from the Atlantic to the Pacific and was renamed Douglas. In 1988, Hurricane Joan made the crossing to the Pacific, where it was renamed Tropical Storm Miriam.

Can a hurricane have three names?

You bet. In 1961 Hurricane Hattie devastated Central America. Then it jumped to the Pacific where it was called Simone. Then it bounced back to the Atlantic side via Mexico and was named Inga.

Right now, the remnant of Hurricane Dean is slowly spinning off the coast of Santa Barbara, CA, and pumping moistures into southern CA, Nevada, Utah, northern AZ, Colorado, and even Wyoming.