I can't remember whether I heard this, made it up, or received it in a visitation from some linguistic gnome ~ take your pick, and feel free to tell me if I'm way off base.

Hemi- refers to things that are divided horizontally.
Semi- refers to things that are divided vertically.
Demi- refers to a smaller subset or version of the original.

Hmmm. Now that I type it out, it definitely looks like the work of gnomes!