Two Rooks are stronger than one Queen. Let's assume that the material is equal, except that one player has a pair of Rooks while the other player has a Queen. In what situations is the Queen stronger than the pair of Rooks?
Since everything else is symmetrical, she can be stronger only if two rooks are not well coordinated and there is a presence of pawns on both wings. This gives you a chance to quickly snatch a few pawns and set them in motion, which should tie at least one rook.
If you fail to utilize this, then rooks will get coordinated-one will defend while the other will attack-and will slowly but surely regroup to attack the king, after which they can exchange themselves for a queen and a pawn resulting in a won endgame. Something like this:
This is just an example, of course, but that is how I would play it. It all depends of the position but I doubt you can stop rooks from regrouping like in the above diagram.
Hopefully we shall see other interesting answers as well, looking forward to it.
EDITED IN RESPONSE TO OPs COMMENT:
It doesn't matter, the endgame should be won for the side with the rooks anyway.
The only "saving grace" for the side with the queen is asymmetric distribution of the pawns, plus they should be far ahead so when side with rooks enter pawn ending, side with the queen can still create a passed pawn. I guess it all depends from the efficiency of rooks regrouping. I am pretty sure they will be able to tie the queen down eventually, after which the transposition into a won endgame will occur.
To answer this question it would be important to find the exact evaluation of the following position:
This is the most realistic case I can come up with. If we can prove that Black can win with the regrouping of his rooks as above, then it is safe to assume Black wins in equal positions with 2 rooks vs queen. I will analyze this myself, or even post this as a question.
No mater how I alter the pawn structure, White holds the game by sacrificing king-side pawns to open my king.
Then he uses queens mobility to attack my queen-side pawns and to harass my king.
My rooks can not regroup into winning lineup because of the threat of double attack ( check on the king + attack on the rook/pawn ) or simply because side with the queen has perpetual check.
The only way for queen to win is to strip opponents king naked and use double attacks to win material. Even with such a pathetic setup for rooks as above they can still hold the game.
To win with the queen, you will need weaknesses in opponents pawn structure and you will have to blow up opposing kings cover.
Hopefully you will be able to exploit pawn weaknesses and bad rook coordination with a double attack to win material.
Using queens mobility and giving "smart" checks is the key here. Keeping as many lines closed can help you-avoid pawn exchanges to restrict rooks mobility and to have greater chances of snatching a pawn with double attack.
The side with the Queen will be stronger when the player with the two Rooks has a lot of weak pawns and/or squares. Yet, the balance in a Queen versus two Rooks situation depends a lot on the position. Thus, a rule of thumb is to prefer the Queen when it has a lot of weak squares and pawns to attack.