I think Rouge was meant to be a counterpart of Knuckles much like Shadow is a counterpart of Sonic. She debuted in Sonic Adventure 2 on the opposite team of Knuckles, and had many abilities similar to Knuckles' like gliding, clinging to walls, and digging. She also had a fascination with treasure, and became Knuckles' rival after he found out she wanted his Emerald.
To Rouge's dismay, Knuckles shattered the Master Emerald in Sonic Adventure 2 to keep Eggman from stealing it. Rouge and Knuckles both ended up looking for the pieces, and Knuckles eventually had to track Rouge down to acquire the pieces she had found. This led them to a showdown that ended in Rouge slipping on a narrow beam and Knuckles saving her from falling in lava. The two slowly gazed at each other, and then Rouge insulted Knuckles before giving him her pieces of the Master Emerald. The two have had a flirtatious, hate-your-guts kind of relationship since.
I think Rouge and Knuckles admire, but infuriate each other. Neither wants the other to die, or would wish the other harm, but both want the Master Emerald, and the Master Emerald will forever be between them. Even when Knuckles thinks Rouge has given up, Rouge tries to take the Emerald again.
Looking past the Master Emerald, I think Rouge feels in-debted to Knuckles for saving her life, and when given the opportunity to return the favor, she saves his in Sonic Rivals. They both owe each other their lives.
There's an obvious romantic tension between these two. Rouge definitely has a crush on Knuckles, but she keeps it a secret, and Knuckles has yet to realize it. Will Sega continue the love saga between echnidna and bat, though? I personally don't think the two have a romantic future, because the Emerald will always be in the way. Rouge can't control her greed, and Knuckles has pledged his life to guarding the Emerald.