This school of thought was developed in the late 90's--near the beginning of the Pokemon anime TV series--and pretty much was a collective epiphany for viewers of Pokemon and lovers of Team Rocker everywhere. Therefore, there was no real founding father/mother of this concept; as many Pokemon-analyists and general Team Rocket devotees all eventually stumbled on this possibility by their own means. The following theory guide is a little bit tongue-in-cheek, but true. So true.

(We know they're at LEAST sleeping together...)

Rocketshipping is the theory that centers around all or any one of the following thought processes and/or combinations. Most Rocketshippers believe in at least one of these; and many employ the use of many while writing fanfiction. We, the Rocketshipping population of the world, promise to defend and encourage this alleged relationship until the end of time, while refuting mercilessly any evidence mounted against us and disregarding the possibility of our own fallacy.

The Rocketshipper's Theory Guide:

Article I of the Rocketshipper's Theory Guide:
Sections:
1. Jessie and James are in deeply, madly, truly in love. This is a fairly straight forward and obvious provision. It is the most common and widely accepted provision of this theory. It entails the following principles:
   i. They may or may not be in a sexual or casual relationship.
   ii. They may or may not have accepted this alleged love.
   iii. They may or may not have confessed this alleged love.
   iv. They may or may not have confessed this alleged love, and may or may not be hiding  it from the series' other key characters as well as the audience.

2. Jessie and James are engaging in sexual or intimate activities. This includes anything slightly more intimate than their traditional fear-hug. This provision is usually reveled in by some of the kinkier members of the Rocketshipping population, though not always accepted by supporters of Section 1. It entails the following principles:
    i. They may or may not be in love (blatant contradiction of Section 1, this provision says although Jessie and James may have some emotional feelings for each other, the most powerful feelings in the relationship are that of an amorous nature.)
   ii. Their lust is a combination of pent-up feelings. (this provision holds that Jessie and James are in love, but they can't find it in themselves to tell the other; so they just have sex instead.)
   iii. Passionate love equals passionate sex. (this refers basically to the fact that Jessie and James are deeply in love and their lust stems directly from that. Commonly accepted by supporters of Section 1.)
   iv. Secret sin. (their sexual/quasi-sexual activities are kept hidden from all other characters in the series. This may or may not include Meowth).

3. Unrequited love. This section deals with the fact that one party has feelings for the other, and they are unreciprocated or blatantly ignored, for assorted reasons:
   i. Jessie loves James, but he does not love her, based on his homosexuality, (the most common defense against Rocketshipping is James' alleged homosexual affiliation.) lack of commitment (since being scarred for life by Jessibelle, his fiancee), or various reasons.
   ii. James loves Jessie, but she does not love him, based on various reasons.
  iii. Jessie confesses her love, and though James feels the same way, he rejects her for various reasons.
  iv. James confesses his love, but though Jessie feels the same way, she rejects him for various reasons.

Article II of the Rocketshipper's Theory Guide:
Sections:
1. Common possible reasons of why Jessie and James will not or are not getting together. Rocketshippers can acknowledge the other side of the argument despite the fact that they are aware the opposition is dead wrong and neck-high in self-delusion.
   i. Homosexual tendencies, relating specifically to James's effeminate personality and penchant for cross-dressing.
   ii. Conflicting personalities.
  iii. Jessie and James's role of being best friends is strictly platonic: the degree of closeness that has been acquired can never be shifted to a romantic gear.
   iv. Other interests: Jessie and James have someone else on their minds.

2. Evidence. This section is made up of the most common pieces of information used by Rocketshippers to defend Jessie and James' relationship.
   i. James's attraction to certain one-time characters that appear in the anime; specifically, the Ghost of the Maiden's Peak. When Jessie rescues James from the Ghost, after learning in disbelief that he doesn't want to go with the Ghost, James proclaims, "Jessie! You really DO care!"
   ii. Certain episodes have illustrated Jessie and James's caring towards each other in ways that seem more than platonic. Examples would be during the St. Anne saga in which James and Jessie share a tender (if humorous) moment; James's kind actions at the end of the "Princess VS. Princess" episode in which he goes out of his way to make Jessie happy despite her crushing defeat; James's disposition towards making up for Jessie's loss in "The Ultimate Test," and finally (the 'big guns'), "Holy Matrimony," in which James gives up his life of riches and luxury (and dominatrix fiancées) to return to a life of crime and low-income with Jessie and Meowth (though he shares a particularly tender moment with Jessie in the hot air balloon that ends with the two of them holding hands, despite the fact that most anti-shippers declare they are simply "shaking" hands.)
   iii. The official Pokemon manga depicts Jessie and James as a couple; wedding bands adorn both their fingers and Jessie's abdomen is swelled with the signs of pregnancy. (Her hair is also cut short, which is unrelated but note-worthy just the same.) This is the single-most used image amongst Rocketshippers, and needs no introduction. However, it is worth mentioning that the manga follows a different story than the series. 
   iv. Most Rocketshippers also deem it necessary to mention most of Jessie and James's unnecessary touching, including fear-hugs and just general excessive groping. In fact, Rocketshippers take it upon themselves to document all of Jessie and James's encounters of fondling.

3. Common fanfiction approaches. Whilst throwing Jessie and James completely out of character, fanfiction authors seem to have a steady approach to Jessie and James's hook-up process:
   i. The accident. Some other trivial character (such as Meowth, Misty, Ash or Brock) lets it slip that either Jessie or James is in love with the other. Hilarity and/or drama ensues, and finally it is established that the other reciprocates those feelings.
   ii. The guilt trip. Something happens to either Jessie or James, and in his/her absence the character left behind wallows in self-pity and wonders why he/she never told the other character the truth.
   iii. The heroic escapade. Jessie/James is kidnapped or turns up missing and it falls on the remaining member to fly to his/her rescue. Of course, they fall in love in the process.
   iv. The tragedy. Finally they reveal their feelings to each other, only to have something appropriately tragic befall their short-lived bliss. This includes but is not limited to, the death of one or the other, the death of Meowth, one of them falling into a coma, etc etc.
   v. The betrayal. One of the characters is in the process of hooking up with someone else, not realizing the other character's undying love for him/her. This, of course, creates high amounts of drama and ends in either sex or suicide.
   vi. The sappy retch-fest. Only for die-hard romantics, this type of story usually deals with high amounts of cuteness and is usually told in a first person perspective. Includes gratuitous amounts of mush and the conclusion is accompanied by a dreamy smile and a tear in one eye.

4. The Butch and Cassidy Clause: This section basically is to clarify that being a Rocketshipper does NOT require support for the Cassidy/Butch relationship. This, in fact, is a lesser-known phenomena known as Neoshipping, and is generally less supported than Rocketshipping due to Cassidy and Butch's secondary character status. As well as bad hair and rip-off motto.

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the end.
I'll add more as I think of it. I think I covered the subject pretty well. Bottom line: Rocketshippers want Jessie and James to get FREAKKKY! But because we're romantics, not horny perverts.