^{This is a repost of a question I posted at MSE.}

Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus:

• There are five clear-cut families of geodesics.
• Most of the geodesics are "ergodic": aperiodic and covering either the entire surface - by spiraling endlessly around - or substantial parts of it.
• Some of the geodesics are "boring": the meridians, the inner and the outer equator
• A few of them are "æsthetically pleasing": returning to their starting point after just a few circuits.

Does the structure of geodesics change when twisting the "hose" before gluing its ends?

E.g., there might be no equators anymore because after twisting the (two) equators lost their "ends".