Invader Zim (TV series)
The relationship between Gaz and Zim in the series could be accurately described as unfriendly, and aloof at best. Throughout the course of the show, neither character has ever expressed anything close to affection for, or even a friendly demeanor towards, each other.
So far, Gaz has hardly spoken more than a few lines to Zim and seems to consider him a vague nuisance, rather than an actual threat to Earth. In-between the events of the first few episodes, she and Zim met, though it can be said that it was a first meeting that was far from friendly or affectionate.
However, even though they don't interact much, they do have some things in common: namely, their mutual hatred of Dib, and their mutual desire to cause him great pain in various forms (though they have never succeeded in killing him to date). They even have a loathing for Earth itself. Also, both Zim and Gaz have a penchant for being melodramatic when angered, as well as a love for destruction. Both can also be very determined in getting whatever it is they want and can be rather stubborn.
They also share common interests, such as a surprisingly high tolerance for GIR and his hyperactivity, interestingly Gaz also has a knack for robotics like Zim since she was the only human who has been seen creating security robots that can be quite deadly as demonstrated in "Gaz, Taster of Pork".
Their contact throughout the series is, unfortunately, very limited. Not unlike Dib's disregard for GIR, Zim often brushes his mortal enemy's little sister off when dealing with the pair of them. It is hinted in "Future Dib" that he thinks she's "scary" and might be intimidated by her.
This fact becomes even more apparent later in the same episode, when the Zim-controlled Dib Robot-Spy appears to have a spastic fit in front of Gaz, enraging her and Zim screams when he sees her coming "his way" and ordered the robot Dib to run away despite the fact that Zim was in no real danger.
In the original pilot, Gaz seemed to show a bit more interest in Zim than she does in the main series. She smugly smiles at him when he is about to chew beans, and looks down at him when he is in pain from the allergic reaction. In the unused script for the pilot, Gaz drags Zim back to his house after his allergic reaction. Gaz explains that she helped him to get on her brother's nerves and even provides intel on Dib's surprise food fight.
In "NanoZIM", Gaz beat up Zim in what she thought was an online video game. It is notable that she recognized Zim's voice. However, his voice is pretty easy to distinguish, and she's probably used to hearing it, considering all the fighting between Zim and Dib.
In "The Wettening", Zim held Gaz above his head to protect himself from the rain (completely ignoring the umbrella that was in her hand, and ignoring her threats). Gaz probably would have beaten him up if Dib didn't intervene with his puddle splash. This is one of the few instances when Gaz did not follow through on her threats.
In "Planet Jackers", Gaz didn't seem to care that Zim was in her house but rather was annoyed about Dib yelling about it.
In "Bloaty's Pizza Hog", Dib asked Gaz if she even cares that Zim's trying to destroy mankind, but she flatly stated how: "he's so bad at it." this is despite the fact that Zim has the very real capability to obliterate the human race, which suggests that unlike Dib, she can tell Zim can't use the technology he has properly.
It should also be noted that this episode demonstrates how Zim underestimates Gaz's abilities, and her determination to rescue her brother. This is proven when Zim triumphantly told Gaz he expected to be amused by her "pitiful" attempts to get Dib out of an "impenetrable" containment tank that Zim made himself. Gaz promptly kicked the tank lightly, causing the entire thing to shatter and flood the room.
Later, Zim also showed outright condescension toward Gaz: when he cuts the power to the teleporters, Zim referred to her as "Little Gaz," reflecting the fact that his species judge others by height.
Despite Zim's demeaning views of his nemesis' sister, Gaz proved herself to be more than superior to the maniacal Irken during her escape from Zim's Space Station. During the said escape, Gaz displayed her prodigious piloting skills and was even shown to be capable of flying literally cockpit-to-cockpit with Zim's ship, even sparing him a smirk and mocking wave in the process.
In "Rise of the Zitboy", Zim tried to hypnotize Gaz with Pustulio, but she was reading a book and didn't pay much heed to Zim's presence. However, after asking if she was sure she didn't feel compelled to gaze upon Pustulio and being answered with a threatening growl from Gaz, he acquiesced to her wishes. This was probably due to Zim seeing for once that there was no point in trying to hypnotize her and better to leave her be, though he did seem annoyed she was ignoring him, which most likely was something his ego couldn't accept.
In "Tak: The Hideous New Girl", Gaz claims that Zim's voice is "making her sick", and then sprays him with soda (however, it should also be noted she then sprayed Dib as well when he tried to thank her).
Later when Zim takes her and Dib to his base, he threatens to "melt Gaz's face off or something" if she touches anything. Interestingly, he leaves Gaz in the base without any restraints while he goes to stop Tak's plan, probably because he is aware that unlike Dib, Gaz presumably doesn't care enough to interfere with his mission.
Later, after Dib gave her the curse of the Shadowhog and couldn't find a way to cure her, Dib was so desperate that he asked Zim for help; unsurprisingly, the Irken refused to help fix Gaz's ailment, though the fact that Dib was asking for the assistance probably had a lot more to do with this than his feelings towards her. Later in the episode, Gaz tells the Shadowhog how she knows that Zim is an alien, but how she also believes he is too dumb to take over the Earth.
In "Future Dib", when Zim is controlling the Dib Robot-Spy, he says "Of course, my scary sister. I obey.", while clasping his hands and smiling sweetly. What's interesting is that his creation doesn't repeat these movements, instead opting to place a hand over his heart as if promising something.
In "Dib's Wonderful Life of Doom", right after Dib throws a muffin at Zim's head, she is visibly shocked at what has happened, calling it "horrible" and turning her head away, looking visibly disgusted. However, after Zim hysterically runs out of the cafeteria, she says that Zim's hysteria during his escape was "kinda funny".
In "Parent Teacher Night", after Zim tapped the bottom of Dib's cup up to splash punch into his face she chuckled, and when Dib poured punch onto Zim's head she snickered again.
In the unmade episode "Mopiness of Doom", Gaz apparently wasn't at all bothered by Zim disappearing from Skool for a few weeks; in fact, she was much more concerned with how bizarre Dib was acting.
In "Lice", however, even though Gaz didn't come in any physical contact with Zim, she did call him a whiner whenever he - or in fact, anyone else - yelled in pain.
In "The Voting of the Doomed", Gaz casts her ballot for Willy when the skool election day arrives. But when the monitors were sensing Zim's popularity level, Gaz's meter was blinking green, meaning that she would have voted for Zim instead (Ms. Bitters actually mentions this herself when explaining how the indicators work, saying that this means that someone may find interest in what the other candidate has to say). Additionally, after he gives his "salted nuts" speech, Gaz cheers along with the other kids but is not seen wearing a brain monitor.
It should be noted that indirectly at least Gaz has saved Zim's life three times, first in "Battle-Dib" where Gaz was seeking revenge on Dib for stealing and eating the last pizza and couldn't give his presentation on Zim to The Swollen Eyeball Network.
The second time was in "Mysterious Mysteries" where she was asked if her brother revealed the possibility of "intelligent extraterrestrial life" unsurprisingly her testimony negatively depicts Dib and Zim babbling stupidly, followed by Gaz knocking down Dib with a sweep kick, showing how she sees them as idiots and at the same time discrediting her brother on national TV.
The third and final time was In the unmade episode "Ten Minutes to Doom", Gaz only grudgingly helps Zim to retrieve his PAK from Dib. Note that, despite the fact that Zim will die without his PAK, she only decides to help him when he tricks her into thinking Dib also stole her Game Slave 2. Later in the episode, after Gaz saves Zim's life unknowingly, upon discovering that Zim was the one who actually stole it (in an attempt to convince her to save his life of course), Gaz chases after him with the likely intent of physically or verbally abusing him.
As shown in Invader Zim: Enter the Florpus, despite not caring about Zim's repeated attempts to take over the Earth, Gaz is willing to aid her brother in defeating Zim if his threat is indeed genuine, such as what happens when Zim uses Professor Membrane's Membracelets to teleport the Earth across space to impress the Almighty Tallest (creating a Florpus Hole that threatens the world as a side effect) after sending Professor Membrane to Moo-Ping 10 and replacing him with Clembrane.
Aside from joining forces with Dib to stop Zim from destroying the world, Gaz truly hated Zim for abducting her father, demonstrating that she tolerates Zim but that there are certain boundaries that she isn't willing to let him cross.
Invader Zim (comic series)
Issue 16 is the first issue to show any significant interaction between the two characters. Zim agrees to take Gaz to Arcadikon, after she grows depressed at having beaten every video game on Earth, in exchange for advice on how to scare Dib. He later states that he enjoys talking to her, and gets annoyed when she says they don't do it more often because she hates him, responding that he "hates her more".
Notably, Zim also spends the whole issue calling her "Gus", indicating that he doesn't actually know her name, which is similar to the animated series, where Zim rarely called her by name, sometimes calling her "Dib-Sister". It's odd, however, because it seems that he does indeed know her name since he did shout "Gaz!" at a bush when he mistook it for Dib's sister in Issue 1, along with the fact that he called her "Little Gaz" in "Bloaty's Pizza Hog".
In Issue 21, an accident with one of Zim's machines causes him and Gaz to switch bodies with each other. One consequence of this is that they start expressing aspects of each other's personalities - Zim becomes apathetic and obsessed with gaming, much to his own disgust, while Gaz gains Zim's drive to conquer Earth.
She bets Zim that she could do a better job of it than him, with the loser to be banished to the moon. Zim smugly agrees, thinking that Gaz can't win, but in a matter of days, she has the leaders of the world begging her to take over. While initially impressed, narcissistically claiming that Gaz must have stolen his ideas, Zim quickly becomes jealous and confronts Gaz, who merely states that she's stealing his identity before she assumes control of the world. Her first order as ruler is to banish Zim to the moon.
In Issue 40, it was revealed that across the multiverse, in universes where both Zim and Dib exist, they always end up meeting and are destined to be enemies. Their relationship in each universe is unknown. In the "Nice" verse, however, Gaz happily gives Zim a gun, while Zim is visibly apprehensive to accept the gift as he says "I love this, sparkly human".
In the DVD commentary for the episode "Tak: The Hideous New Girl", it was referenced that Gaz might be Zim's love interest (which was most likely a joke made at the time, since Jhonen Vasquez once stated that there are no official romances in the series).
But, at the time the commentary was made, it only served the fans to often pair them up as a couple (Called ZaGr -Meaning Zim And Gaz Relationship to fans). Like all pairings in the series, this is usually done as a joke, or for entertainment purposes.
There is absolutely no evidence of "ZaGr" in the series as is, considering their limited interaction, not even the vague glances and moments that fans often make into evidence for a pairing. It is likely that the pairing itself sprang only from the aforementioned off-handed director's comment.
For years, there were rumors on the internet that Gaz would have secretly sided with Zim against Dib's forces in the non-canon unfinished series finale "Invader Dib", and would join him in ruling Irk later on; however, these rumors were debunked by Eric Trueheart in The Medium-Sized Book of Zim Scripts: Vol 1: Pigs 'n' Waffles. "There was no planned ending", he claims. "Sorry, kids, this rumor is not true. There were never any plans for "Invader Dib."
Also, Melissa Fahn confirmed that Gaz can feel love while Jhonen Vasquez similarly confirmed that Irkens are capable of love as well, but due to the age difference (Zim is clearly a young adult in Irk years, estimated at 160 Earth years old; though not officially confirmed) while Gaz is a child (presumably around 11 Earth years old) it most likely wouldn’t happen.