Can I use a reaction card immediately at the start of the enemy turn?

No, you cannot use a reaction card simply because your opponent's turn has begun.

According to RiftJudge FAQ #10123 and RiftJudge FAQ #9538, you do not receive priority at the beginning of a turn or phase. Priority is only granted to you when your opponent performs an action that creates a chain (such as playing a spell or triggering an ability that uses the chain) or during a showdown.

Here is the breakdown of why based on the game rules:

1. No Priority at Turn Start: Simply starting a turn or phase does not grant you a window to play cards.
2. Playing Permanents: If your opponent plays a unit or gear at the start of their turn, these permanents resolve immediately. As confirmed by RiftJudge FAQ #5909, playing a permanent does not pass priority on the chain, so there is no window to react to them being played.
3. When You Can React: You can only play a reaction card if:
   • Your opponent plays a spell (which creates a chain).
   • Your opponent triggers an ability that uses the chain (such as a "When you play me" ability).
   • A showdown is occurring.

In summary, you must wait for your opponent to perform an action that specifically places an item on the chain before you can play a reaction.