The Flat Brim: Reyes has a legacy in St. Louis, where the Cardinals are among the most flat-brimmed teams in the league. Supposedly, Reyes wore the flat brim because a curved one hit his glasses back in the day. Now, it's a fashion statement, although there's probably some practical application as well.
Bonus: Better view of ears and upper sideburns.
The question is how long the flat brim will stay in vogue and whether Reyes will have the fortitude to stick with it long after the trend has been forgotten. I dare say he will, even if it makes him look like a teenager wandering around the mall. In fact, that might help batters underestimate Reyes, so it's a good thing.
Hologram Underbrim: Fausto Carmona does not remove the holograms from the underbrim of his hats. I've never heard a good explanation for that, other than him intimating it's just a ritual or something. Like the flat brim, the holograms are also a cultural fashion phenomenon you can witness walking down just about any street. On the baseball field, I think they're particularly useful for day games, when there's a glimmer of a chance that a ray of sunlight will bounce off the shiny little sticker and temporarily blind a batter.
Edge: Holograms on the underbrim. Sneaky and less conspicuous.