1. I've always viewed both skateboarding and jazz as sort of subversive, anti-establishment activities. Within the history of both, there are varying levels of how "underground" or popular the activity is at different times. When I skated, there was only one other kid in school who did it. When I was learning to improvise years later, there were two. When you traveled to different towns, you could always kind of tell who a skater was by their shoes, or a jazzer by a Mingus T-shirt or if they were holding a copy of the Miles autobiography.
2. Both jazz and skateboarding deal with a basic language and variation on that language. Grossly oversimplified, jazz uses scales, arpeggios, musical fragments and licks, while skateboarding uses the ollie, boardslide, and grind. (Of course there is way more to both than that!) On an an extremely basic level, a "line" might be a series of tricks or a series of licks.
3. Obstacles: One way variation can happen in both activities is by changing context. In jazz we use the basic language over forms of varying levels of complexity, while skateboarding applies the same maneuvers and variations over ramps, rails, curbs, "flat ground" or whatever else the skater can think of.
4. Related to obstacles, both skating and jazz are a reaction to one's environment. Much of jazz has been a reworking of or a reaction to the pop music of the day - even finding new uses for old tunes. Skating finds new uses for the architecture around us. Jeff Grosso and Ed Tempelton talk more about this here:
5. Style: Each skater or musician chooses what part of the language they will master and utilize, how they will vary it and what context they will put it in. These factors contribute to an individual's style. Sound and musical inflection contribute to musical style as well, while specific body movement contributes in skateboarding.
6. The history of both activities contains explosions in individual's ability to handle more complex material - four-minute mile moments where the impossible becomes possible, where things only professionals can do become basic maneuvers for the next generation.
7. Session Culture: The little weirdnesses in sessions (jazz "jam" sessions vs. skate sessions at a park) are surprisingly similar. There is an etiquette to the order in which someone drops into a ramp as there is a typical solo order in jazz. Courtesy is important as one is careful to not use an obstacle too long or play too many choruses on a tune. Some show off. Artists are always sizing each other up - "is he better than me?" Most people are there to have fun and be inspired.
8. Christian Hosoi and Tony Hawk are the Lester Young and Coleman Hawkins of Skateboarding. Tony Alva is the Louis Armstrong and Rodney Mullen is definitely the Charlie Parker. If you do both activities let me know if you agree.
9. Camps: Skateboarders argue about being a vert, street, or freestyle skater in a similar way to jazz musicians arguing about being a "staight-ahead" or "free" player. Battles are fought but never won in both.
10. Gear: Innovation in gear (instruments, recording equipment, obstacles and skateboards) has affected every aspect of both activities, with a major innovation often leading to an explosion in complexity. In both activities, there are those who focus way too much on gear, and should probably just practice.
11. World view: Both skateboarding and jazz change one's view of the world forever, whether the person is currently engaging in the activity or not. A handrail is always something that could be slid just as a pop tune is always something that could be analyzed and improvised over - subject to the imagination of the beholder.
12. Art vs. Sport, technique vs. style: There are forever battles in both activities over whether it's better to be technical or artistic (Hawk/Hosoi, Hawkins/Young). In skateboarding they argue if it is an art or a sport, while jazz musicians are arguing about having a "voice" versus the ability to play any style. Camps form, battles ensue... Nobody wins but it keeps people engaged.
Here is Mark Gonzales, the godfather of street skating, doing his thing to John Coltrane: