Linzin

Linzin is a Het ship between Tenzin and Lin Beifong in The Legend of Korra series.

Canon
Tenzin and Lin dated at some point in the past. They have known each other since they were children due to the fact that their parents were good friends. At some point, Tenzin and Lin started to grow apart and their romantic relationship began to crumble apart because they had different goals in life. Once a woman named Pema confessed her feelings for Tenzin, he decided to break up with Lin so that he could date Pema. Lin was very upset and did something to Air Temple Island because of her heartbreak. She also attempted to arrest and imprison Pema for stealing Tenzin from her.

As time passed, Tenzin and Pema eventually got married and had four children together (two sons and two daughters)--Lin, on the hand, never had kids and ultimately chose to stay single. When Korra arrived in Republic City, Lin was initially bitter towards her because of her past relationship with Tenzin. When Lin decided to have the police force guard the arena after a threat from Amon, Tenzin convinced her to allow him to join her and she smiled that it would be like old times. Tenzin also convinced Lin to be nicer to Korra and give her a chance.

Tenzin and Lin later worked together to investigate Hiroshi Sato when he was found to be an Equalist. They were caught in a trap and rescued but all of the cops working for Lin were taken away. Lin was depressed and decided to resign as chief and Tenzin told her that she needed to rest. When Korra was kidnapped, Lin left the hospital and Tenzin told her that she should still be resting. After Korra was rescued, the Equalists began to attack Republic City and Tenzin entrusted Lin to watch his family while he was away. After Pema had her baby and Tenzin returned, the airbenders decided to flee and Lin went with them to protect them. While the Equalists were chasing them, Lin ran behind to distract them and Meelo said that she was a hero, which Tenzin agreed with. Lin was eventually captured and had her bending taken away as a result of her sacrifice.

Over six months later, Lin heard a crash inside the police station and went to see what happened. Mako explained that he broke up with Korra and Lin remarked that the damage was nothing compared to what she did to Air Temple Island after Tenzin broke up with her. After the Red Lotus escaped from prison, Lin went to Ba Sing Se to inform Tenzin and explained that they needed to protect Korra. Lin eventually went to Zaofu where she confronted her half-sister, Suyin. Su remarked that she could see why Tenzin dumped her which provoked Lin into attacking her.

Fanon
Linzin is a popular ship in the Legend of Korra fandom. Due to way that Tenzin and Lin behaved around each other, many fans speculated that they had dated in the past. This especially became the case when it was revealed that Tenzin dated somebody else before Pema. Some fans hoped that this was not case as they felt that it would disprove Tokka since Tenzin and Lin would be cousins. However, it turned out that Tenzin and Lin did date in the past. Many were excited and began to ship Linzin while others were disappointed that Tokka was disproved, as it was not yet known that Lin had a half-sister.

A lot of fanon was written about Linzin as soon as they were revealed to be a couple in the past. Many fanon writers wrote about how the break-up happened. A lot of fans speculated that Lin did not want to have children while Tenzin pretty much needed to or else the airbenders would die out. Some also made fan art of Toph teasing Lin and calling Tenzin "Twinkle Toes Jr." while embarrassing the both of them. Some fans hated Pema for stealing Tenzin and hoped that she would either die of childbirth with her fourth child or turn out to be an Equalist so that Linzin would still have a chance. However, many Linzin fans also accept or also ship Pemzin.

Trivia

 * It is unknown exactly what Lin did to Air Temple Island after Tenzin broke up with her.
 * According to the show's creators, at least part of what hurt Lin and Tenzin's relationship was the fact that Tenzin wanted to have children while Lin didn't.