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 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 Game Slave, which was placed into her body courtesy of a certain alien). And in Gaz, Taster of Pork, after coming clean about his original intentions and fearing his sister has failed her test (which in fact she passed), he begs the hog beast tospare her and punish him instead. Even though it turns out there's really no need for either of them to suffer, Dib is punished and recieves no mercy for his sacrifice - especially from Gaz.
In Dib's Wonderful Life of Doom, Gaz and Dib are seen quarrelling just like any other pair of siblings would. After hearing from her father about Dib having super powers, she says in a monotone voice, "I guess I'm sorry for being so mean to all those times". However, this was only a dreamlike illusion created by Zim in the hopes of finding out if Dib was the one responsible for throwing a muffin at him during lunchtime earlier; she also, after aging, appears to help him take down the Irken Armada, though again this is only an illusion; in the unfinished series' finale, Gaz would have secretly sided with Zim and his people instead, much to Dib's shock and horror (according to writer Eric Trueheart).
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.
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 pretty much that each other has; seeing that Dib does not have any friends, and Gaz is usually seen alone (unknown if she has friends). They seem to always sit together at the same table at lunchtime.
Despite her annoyance with Dib's obsessive nature and her generally mean behavior toward him, her actions in Mopiness of Doom show that she is unhappy when he stops pestering her about the paranormal, going so far as entering his room with a drawing she made about that night's Mysterious Mysteries episode to entice him into coming downstairs, stating that Dib should be "annoying" her right now. While she might simply be irritated by the sudden change altering her routine, she also sees that Dib isn't truly happy when he isn't pursuing his real interests and perhaps even feels lonely or ignored without her brother rambling on at her. Although Gaz is mean to her brother and enjoys torturing him and causing him pain, she would also defend him if someone else caused him to go beyond an acceptable level of Dib suffering. 
Invader Zim (comic series)
Jhonen has stated that one of the things he wants to do with the comic series is expand on Dib and Gaz's relationship, presenting them more as siblings. In the first issue, Gaz is shown annoyed and disgusted by how out of shape Dib's become, and by his singing of his own workout montage music while trying to get back in shape; Gaz provides the music instead, just to help get him back to being less disgusting.
In Issue 3, Gaz mocks Dib's obsession with Bigfeet by revealing she's already met him and taken pictures with him that are now posted online. After making fun of Dib's reaction, Gaz mentions that Zim is on TV posing as an artist. When Dib decides to investigate Zim's newest plan, Gaz pretends to be excited about joining him, but then reveals she's just messing with him.
In Issue 5, Dib shuts off Gaz's online game to force her to spy on Zim with him. In retaliation, Gaz recruits their father's help to find an alternate universe where gamers rule and non-gamers are enslaved as NPCs. After taking over this world, Gaz kidnaps Dib and makes him an NPC, making him suffer horrible pain through countless games until he apologizes for interrupting her games for his own interests. She then releases him and they go home, where they play a video game together with Membrane; but when Dib wins, Gaz calls him a cheater and throws him through a wall.
Gaz only briefly appears in Issue 10, where she accompanies Dib when he goes to spy on Zim's base. However, she makes it clear that she's only tagging along because he bribed her, and when he jumps on the Voot Runner as it's taking flight, she shows no interest, merely walking away and saying that she's going to have their father put Dib's dinner in the refrigerator.
Gaz again only briefly appears in Issue 11, when Zim tries attacking Dib with a robotic cat. When it fails to trigger an allergic reaction (as it lacks dander) and Zim complains about Dib not becoming hideous, Gaz comments that Dib is already hideous.
In Issue 14, Dib was going to explode if he became too sad due to unknowingly ingesting nanoexploder bots. Gaz feels guilty since she was the one responsible for giving him the vial of nanobots to drink. She goes to great lengths to make sure he doesn't become sad, even participating in his paranormal investigations even though it's obvious that she does not enjoy it. She does this not only to save his life, but also because it's his turn to pick the restaurant for family dinner night and she wants him to pick Bloaty's Pizza Hog.
In Issue 16, Gaz is willing to help Zim terrify Dib in exchange for access to alien games on Arcadikon. However, her efforts are rather halfhearted, indicating that while she doesn't care enough about Dib to not aid Zim, she's also not totally willing to help harm him.
In one of Dib's stories in Issue 17, when Gaz sees that Dib has brought pizza for all the Skoolchildren, she proudly announces to everyone that he is her brother and calls him "the best". However, it is possible that Dib made the story up.