Can I choose to recycle runes at any point during my or my opponent's turn even without having to pay to play something? For a concrete example, if I don't have anything to play but I want to get the Sivir +2 Might from recycling 2 runes on my opponent's turn, can I just recycle two runes?

Yes, you can absolutely recycle runes at any time you have priority, even if you are not currently paying for a card or ability.

According to Rule 163.2.b and the RiftJudge FAQ #404, recycling a rune is a [Reaction] ability that adds power to your resource pool. Because it is a [Reaction] ability, you can activate it whenever you have priority.

Regarding your concrete example:

• Yes, you can do this. You do not need to be in the middle of playing a card to use this ability.
• The timing: Since you want to do this on your opponent's turn, you can activate this [Reaction] whenever you have priority. This includes when your opponent passes priority to you, or in response to a spell or ability they have placed on the chain.

Important details to remember:

1. Priority: You must have priority to activate the ability. You cannot do this while a spell or ability is already resolving, or while a player is in the middle of the "Play a Card" process (unless they have passed priority back to you).
2. Resource Persistence: As noted in RiftJudge FAQ #3736 and Rule 166, your rune pool (the energy and power you generated) empties at the end of each player's turn. If you generate power on your opponent's turn, you must spend it during that turn, or it will be lost when the turn ends.
3. Sivir: If you are referring to a card effect that triggers off of recycling (like a hypothetical "whenever you recycle..."), that trigger will go on the chain and resolve normally.

Essentially, you can "float" power at any time you have priority by recycling your runes, provided you use that power before the turn ends!