Contrarily to the Klein bottle, the quadruple Jeener-Klein bottle is an orientable surface (it has two sides). The name of the quadruple (or more generally multiple) Klein bottle refers to its original inventor, P. Jeener, a French artist and copper engraver specialising in mathematically inspired work (see
wikipedia).
The even multiples of the Klein bottle are all orientable. The odd multiples are not orientable. You can have a look to the
Triple Klein Bottle.