"The relationship between Korean martial arts and Japanese martial arts is usually a touchy one. This is because, like the history of so many other things in modern Korea, it is susceptible to what I like to call the “Colonial Death Touch.”

The Colonial Death Touch works like this. Any practice which can be demonstrated to have its origins in the Japanese colonial period, was reborn during the colonial period partly out of inspiration or imitation of some Japanese practice, or was significantly influenced by similar Japanese practices is ruled to be inauthentic. Inauthentic things, of course, cannot be authentically Korean, and thus risk, at the very least, losing its place in the national cultural or historical repository. At most, it can destroy any popularity such practices might enjoy.

The Colonial Death Touch is sometimes delivered by, for example, Japanese nationalists who want to anger their Korean neighbors. However, it is also often used domestically. For example, practitioners of Korean martial art X might claim that they are superior to martial art Y because they are “pure” Korean while martial art Y is soiled by its evil Japanese roots. I’m sure many readers familiar with Korean martial arts can think of some examples of this.

These sorts of exchanges, whoever their participants might be, are silly childish games of nationalist mudslinging. They depend on a simplistic idea of authenticity, a laughable faith in cultural uniqueness, and a conception of the colonial period as cultural and economic black hole out of which only the bright shining light of Korean national resistance can possibly shine.

One martial art that became popular during the colonial period which remained popular in the postwar period is 검도(劍道, J: Kendō) or swordsmanship. In recent years, perhaps partly due to the ever present threat of the colonial death touch, the martial art has undergone some degree of “Koreanization” while other innovations in technique, uniforms, etc. probably are more simply attributable to the evolution of all such arts across time."

And like that, my answer regarding the patriotic use of Korean symbols with Japanese martial arts has been found. But this did get me wondering about Taekwondo, that most Korean of martial arts. While looking for the date of Taekwondo's inclusion as an olympic sport (using the 100% reliable wikipedia), I noticed something about the history of Taekwondo, which I will copy from the article here (and if you want to see some of the controversy that occurs when trying to say anything from modern Korea comes directly from Japan, take a look at the talk page here.)

"During the Japanese occupation (1910-1945), the practice of tae-kyon was banned.^[4] Although the art essentially vanished^[5], some aspects of taekkyeon may have survived through underground teaching and folk custom. As the Japanese colonization established a firm foothold in Korea, the few Koreans who were able to attend Japanese universities were exposed to Okinawan and Japanese martial arts, and some received black belts in these arts. Koreans in China were also exposed to Chinese martial arts.

After World War II and the liberation of Korea, several Kwans (schools) arose. They included: Chung Do Kwan, Moo Duk Kwan, Jidokwan (or Yun Moo Kwan), Chang Moo Kwan, Han Moo Kwan, Oh Do Kwan, Jung Do Kwan, Kang Duk Won, Kyun Hyung do Kwon and Song Moo Kwan. All taught Japanese-influenced systems.^[6] Most were based on Japanese Karate, most notably Shotokan Karate. As these arts began to be taught openly by Koreans who had learned it in Japan, they were taught under such names as kongsudo and tangsudo. According to Steven D. Capener^[7]:

'This process of development can be broadly outlined as follows: Japanese karate called kongsudo or tangsudo was introduced to Korea just after liberation from Japan by Koreans who had learned karate in Japan. Upon returning, these Koreans opened karate gymnasiums promoting what they were teaching as karate, much like the process followed by the early Judo instructors. Well after these schools became established, the need to "Koreanize" was felt. The process of Koreanization consisted of three main aspects. The first was the selection of a new, non-Japanese name. The second was the creation of a system of techniques and training which was distinctly different from that of karate, and the third was the attempt to establish t'aegwondo's existence and development within the historical flow of Korean civilization.'

In 1952, at the height of the Korean War, South Korea held a massive show of patriotism, including a martial arts display where all the Kwans of Korea displayed their skills. Major Nam Tae Hi stole the show when he smashed 13 roof tiles with a forefist punch. Following the demonstration, President Syngman Rhee instructed General Choi Hong Hi to introduce the martial arts to the Korean army.^[11]

By the end of the Korean War, nine schools of martial arts had emerged, and South Korean President Syngman Rhee ordered that the various schools unify under a single system. A governmental body selected a naming committee's submission of "tae-kwon-do". Following the submission of the name "taekwondo" on April 11, 1955 by General Choi Hong Hi, the name was unanimously accepted.^[12]"

If you read the entire entry, it does seem that Korean martial arts have a long history, and that their modern forms are very distinct from modern Japanese forms. However, current Taekwondo may have originally been based on Karate. Despite this, modern taekwondo is vastly different to karate. Taekwondo is as Korean as apple pie is American (or something.) But even mentioning any Japanese influence can lead to arguments with strong supporters of Korean nationalism and of Taekwondo. I personally just find it funny that some places in the States call Taekwondo "Karate" because they believe that it will attract more students.