Twining plants achieve vertical growth by revolving around supports of different sizes on which they exert a pressure. This observation raises many intriguing questions that are addressed within the framework of elastic filamentary structures by modeling the stem close to the apex as a growing elastic rod. The analysis shows that vertical growth is achieved thanks to discrete contact points and regions with continuous contact, that the contact pressure creates tension in the stem as observed experimentally, and that there is a maximal radius of the pole around which a twiner can climb.