Does a copy of a spell with entwine have to pay the entwine cost again?

When you copy a spell, the copy replicates the original spell exactly as it exists on the stack, including all choices already made — such as modes, targets, and any additional costs that were paid (like entwine). This is established in CR 707.10, which states that copying a spell copies all choices made for it.

Entwine (CR 702.40) is a keyword that represents an additional cost you may pay when casting a spell to choose all of its modes instead of just one. That cost-paying decision happens only at cast time. A copy is never cast; it is simply placed on the stack as a new object (CR 707.10a), so there is no casting step where you would be asked to pay costs again.

The copy will already have both modes selected (assuming the original had entwine paid), and it will resolve with all those modes — no extra cost required from the copying player.

Example: Your opponent casts Grab the Reins with its entwine cost paid, choosing both modes (gain control of a creature AND deal damage). You copy that spell with Twincast. Your copy also has both modes and resolves fully — you do not pay any additional mana for the entwine; the modes are already locked in on the copy.

Note: if you are somehow given the choice to change modes or targets on the copy (e.g., via certain copy effects that allow it), that is a separate permission granted by the copying effect itself, not a re-payment of entwine.