Although Tallest Red and Tallest Purple share equal power over the Irken Empire, which exalts height, they do not always get along. Between the two, Tallest Red is shown to be the more competent one (although they both show blatant disregard for any real political responsibility, which is one of the underlying causes for their would-be downfall).
Both are often easily distracted and Red often resorts to beating Purple for entertainment. The latter usually laughs it off, but is also occasionally frustrated by it. The two get along well in general though with their mutual love of snacks, nigh-limitless power, and many jokes at the expense of Zim and other Irkens.
Making their first appearance in "The Nightmare Begins", it becomes clear that Red is the smarter and arguably dominant of the two regents. Ordering Purple to be shot in the eye with lasers and proceeding to grin at his pain establishes him as a bully to his otherwise equal partner. When appearing publicly in front of their subjects, Red often dominates the speeches. In "Battle of the Planets", Red delivers most of the speech to their Planetary Conversion Team. It is established as the series progresses that Red is the most competent as a leader, while Purple is often depicted as childlike and impulsive.
In "Megadoomer", Red reprimands his partner when he complains that he doesn't like the robot they are sending to Invader Tenn, saying "We didn't build it so you could like it". In "Backseat Drivers From Beyond the Stars", Red takes the situation of Massive's loss of control by the horns, checking the ship manually for the source of the problem as well as disposing of their ruined power core. Meanwhile, Purple flails around in a panic while eating donuts frantically. In "Hobo 13", Red covers up Purple's numerous slips of the tongue as they try to convince Zim to attend military training, which they hope will destroy him.
When it comes to making plans, Red is the more adept. They often play tricks on Zim when he calls his leaders, sometimes in order to excuse themselves from an unwanted conversation with the would-be Invader. In "Backseat Drivers From Beyond the Stars", Red fakes being attacked by an "enemy vessel", pushing down Purple in the process. In "Tak: The Hideous New Girl", they build marionettes of themselves and talk to a completely fooled Zim with them. Red later knocks off the head of Purple's puppet, replacing it with Purple's own head and proceeding to whack it with the arms of his puppet. When Red pushes Purple around like this, he doesn't usually fight back. In many ways, Purple is a punching bag for Red.
However, the two Tallests are also known to get along well. Many episodes show them peacefully enjoying their power together, such as in "Walk for Your Lives", which shows them happily "pummeling" the Invaders together.
They both manage to rule their Empire efficiently, despite wasting most of their time. They have a mutual love of snacks and playing around, and their joined hatred of their inferiors bring them together. They take joy in abusing Invaders like Skoodge and Zim, who both happen to be very short.
On occasion, Red can even be seen as being protective over Purple. In "Tak: The Hideous New Girl", Purple is indignant when he learns that Tak's plan of an offering to them was foiled. Red defends the pouting Purple by saying "He likes snacks, Zim", to the otherwise victorious Irken.
Overall, Purple and Red both manage to keep the Irken Empire on its feet, and their relationship consists of both peace and strife.