Let me say that in my personal opinion, the Kukkiwon is the sole source for legitimate Taekwondo and if you are not affiliated with the Kukkiwon, you cannot claim to teach Taekwondo. Why do I say that? Well, Taekwondo is unique in the martial arts community because it's the only art essentially created by a national government rather than by martial artists. In the mid-1950s, the Korean government did not want multiple schools (kwans) teaching their own systems. Rather in this period of rejuvenance following the end of Japanese occupation, the Koreans wanted something that the Korean people could unify around, and numerous martial arts schools scattered across the country teaching what was essentially Japanese Karatedo would not suffice. So on April 11, 1955, there was a meeting among the leaders of the various kwans where the name Taekwondo was officially selected at the suggestion of General Choi Hong Hi, and that's where the simplicity of Taekwondo politics ends.
As Taekwondo's popularity began to grow, the Korean government began development of the Kukkiwon (National Academy) which was completed in 1972. The Kukkiwon was to be the new international governing body of Taekwondo as a whole. Dr. Un Yong Kim was selected as it's first president, mainly because he was also the Vice President of the International Olympic Committee and the Korean government wanted to see Taekwondo included in the Olympics. To accomplish this, in May of 1973 the World Taekwondo Federation was set up to oversee Taekwondo competition (under which the KTA was reorganized as the national governing body). Ironically this is also why Choi moved to Canada to establish the new ITF headquarters. The 1976 Olympics were set to be held in Montreal and Choi planned to gain Olympic recognition for his ITF-style competitions. However, because of Dr. Kim's involvement with the IOC, that was never going to happen and upon hearing of the WTF's selection, Choi immediately began to denounce the idea of Taekwondo as an Olympic sport. Go figure.
Upon establishment of the WTF, the organization was given an office at the Kukkiwon and Dr. Kim was elected it's first president as well. This has led to mass confusion that the WTF and Kukkiwon are the same organization, but they are indeed separate entities who happened to share a building (and a president). The Kukkiwon oversees all Taekwondo training and certifications, while the WTF governs over tournaments on an international level. There is no such thing as "WTF Taekwondo" as an independent style of martial arts. Rather, it is a competition ruleset and nothing more.
This does lead to the some confusion as to where the kwans stand in relation to the Kukkiwon. The kwans have been relegated to fraternal organizations that submit to the authority of the Kukkiwon. The curriculum of the Kukkiwon is the official curriculum of the kwans, however a few such as the Chung Do Kwan and Ji Do Kwan have retained extra material that goes above and beyond what the Kukkiwon requires, thereby maintaining some individuality and kwan identity. Now, the amount of material taught in addition the Kukkiwon curriculum varies as much between dojang of the same kwan as it does across the kwans as a whole.
In Korea, the kwans all defer to the Kukkiwon for official ranking. However, some kwan-specific organizations (mostly outside of Korea) have continued to issue their own documents as well as offer Kukkiwon recognition as an additional benefit. This causes a grey area in terms of legitimacy. As long as the organization as a whole maintains an active relationship with the Kukkiwon, one can argue that the individual members do not need to be certified directly by the Kukkiwon as long as they aren't claiming to be a Kukkiwon school/instructor, and they are still legitimate. I personally believe that if the avenue for Kukkiwon certification is available, it should be pursued. Why risk the question and grey area when it's your credibility on the line? I don't even teach Taekwondo, but you better believe that once my time-in-grade requirements are met, I will be updating my Kukkiwon rank to match my 4th Dan from the American Jidokwan Association.
Where does this leave the ITF? Honestly, this town ain't big enough for the both of us. Bad Western movie references aside, my position still stands. In my experience, the ITF does produce very good quality practitioners who tend to be more self defense based than sport based, which of course is my preference. This is mainly due to the organization's lack of "progress," instead still resembling it's Karatedo origins (except for that weird sine wave thing). However, you cannot argue with the fact that Taekwondo is owned by the Korean government and they say the Kukkiwon's in charge. Now, you may say that because Choi coined the name Taekwondo, when he left with the ITF the Kukkiwon should've chosen a different name for their their art and you're probably right. In fact, there was a period under the KTA when the art was known as Tae Su Do. But ultimately, it was just simpler to continue using the name Taekwondo as there was already an entire generation of students and instructors who had grown up with it. It was a national symbol, and they weren't about to the told what to do by someone who fled the hemisphere. And don't even get me started on how Tang Soo Do fits into all of this...