The constant abuse that Gaz gives to her brother has caused controversy from Christians and a magazine called Christian Parenting Today.
In Dib's Wonderful Life of Doom, when their father, Professor Membrane, says that his DNA seems different than what is normal, Gaz says "I guess I'm sorry about being so mean to you all those times" with a grunt. However, since this was part of an illusion created by Zim, it's unlikely that she would have actually said this. If Dib is in horrible peril and she is forced to help him, she will. But it is seen that Dib isn't always the nicest to Gaz, either.
In NanoZIM, Gaz helps him navigate the micro-suit throughout his body, albeit thinking the whole thing is an online video game and not actually Dib operating a microsuit in an effort to save his brain.In Gaz, Taster of Pork, aside from them holding hands (although the series creator has stated it was an animation error), Dib accidentally casts a spell on Gaz, but feels bad after he finds out the effects. He tries to reverse it to the best of his ability. Meanwhile, though he likely recognizes this on some level, Dib is somewhat obligated to watch over her and rarely seems to be intentionally trying to annoy her. Despite their constant clashes, they will occasionally work together, and there is the rare moment where they seem to get along like family. Dib, however, does sometimes seem to be irritated with Gaz's negativity, such as in Bloaty's Pizza Hog when he responds with a "Uh huh, whatever" when Gaz makes fun of his obsession with saving Earth.
In Dark Harvest, it's shown that Dib cares about Gaz and wants to take care of her when he sees her in the cafeteria and it looks like she's in pain (Even though she was merely playing her GameSlave.)
In Dib's Wonderful Life of Doom, Gaz and Dib are seen quarelling just like any other pair of siblings would. As seen in The Wettening, Gaz seems to be purposefully mistreating her brother, reeling herself in after accidentally letting slip an interest in what he's doing.
However, their relationship is mostly strained. Whenever Gaz is playing on her Game Slave, she doesn't like to be bothered, as demonstrated in the episode Dark Harvest. In Future Dib, Gaz is sent by their father, Professor Membrane, to retrieve her brother. When Zim's Dib-Robot doesn't obey her, she beats it up, which unintentionally saves the day. In Backseat Drivers From Beyond the Stars, Gaz states that "Dib's voice... fills me with a terrible rage!". Their father does little to break up their fighting, especially since he is usually gone.
Gaz isn't exactly on Zim or Dib's side when it comes to Zim's plot for world conquest, as she thinks they are both foolish. In Gaz, Taster of Pork, she tells the Shadowhog that Zim is too "dumb" to take over the world, and that she shouldn't have to listen to Dib's constant ranting about him.
Although they fight often, they do live in the same house, and try to act civil on some occasions. They also seem to be all that each other has, seeing as neither of them don't have any other friends.