Gaz and Zim's Relationship

Zim using Gaz as an unwilling umbrella in the episode The Wettening.

The relationship between Gaz and Zim in the show could be accurately described as unfriendly; aloof at best. Throughout the course of the show, neither character has ever expressed anything close to affection for, or even a friendly demeanour towards each other. 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 and both have 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.

Zim and Gaz during the escape-pod chase.

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, which is not unlikely, considering that Gaz frightens most children and adults she comes into contact with. 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. Despite Zim's very real capability to obliterate the human race, Gaz finds him too "dumb" to take over the world, as she stated in Gaz, Taster of Pork.

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 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 Dark Harvest, Gaz didn't get upset that Zim took her organs, likely only because she was still able to play her Game Slave, and was 'in the zone'.

In The Wettening, Zim held Gaz above his head to protect himself from the rain (completely ignoring the umbrella that was in her hand). Gaz would have beaten him up if Dib didn't intervene with his puddle splash.

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

"Of course, they're right over there."

"he is so bad at it." 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 people judge others by height. However, this has been used by fans as a romantic nickname for her in various fan fiction.

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 said escape, Gaz showed off 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.

Zim attempts to hypnotize Gaz with Pustulio... and fails miserably.

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 and leaves her be whilst beckoning the others to follow him.

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 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.

Zim with Gaz on the elevator.

In Gaz, Taster of Pork, Zim refused to help fix Gaz's ailment, though the fact that Dib was asking for the assistance probably had a lot to do with this. 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.

"That...that was horrible."

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 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. (Miss 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.

In Parent Teacher Night, after Zim tapped the bottom of Dib's cup up to splash punch into his face she chuckles, and when Dib pours punch onto Zim's head she holds in gales of laughter.

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. This is where their secret alliance first forms. 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.

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.

Gaz dreaming of Zim?

In Gaz, Taster of Pork, next to Gaz's bed in her bedroom is a doll that resembles an Irken. It can be seen as the porcine spell is being cast on her.

Relationship

In the director's commentary, it was once referenced that Gaz may be Zim's love interest (which is most likely a joke made at the time, since the creator once stated that there are no official romances in the series), leading 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. 

However in the InvaderCon II, Jhonen Vasquez stated that had he put romance in the show he would have paired up Zim and Gaz together.

In the unfinished series final Invader Dib, there was a theoretical plot that Gaz would have secretly sided with Zim against Dib's forces; it is not considered canon yet because the Invader ZIM movie Invader Dib was never finished.