The inside construction and fingerboard profile of right- or left-handed basses is quite different. If you peek inside the f-hole of an upright bass you'll see the difference.

On the G string (high) side there is a sound post (looks like a wooden dowel) that spans between the top and back of the bass, usually just below but in line with the bridge foot. The E string (low) side has a long piece of wood glued to (or in rare cases, carved as a part of) the top, beneath the E string. These two parts contribute to the sound of the instrument as well as provide structural strength, and unless they are reversed, too, the sound of a lefty-strung bass is not going to be something you'll enjoy.

It takes a good deal of labor and skill to convert a right- to a left-handed bass; it's usually more economical and wise to purchase a bass that has been built as a left-handed instrument from the ground up.