Twining: is a technique in which two wefts cross over each other between warps. There are numerous variations of twining, including variances in the number of wefts, the number of warps crossed by the wefts, and the angle of the warps. (In twining the weft is the active weaving element, and is usually horizontal, while the warp is the foundation element, and is usually vertical.)

You are right, your basket is twined!
Now try to identify the design technique on your basket.