Basically, it's a baseball cap, or any similar sort of hat worn by a chav. The majority of people who wear them are dickheads and the majority of dickheads wear them. A worse variety is a toerag cap, which is one worn backwards.
A Classic Dickhead Cap is a Baseball Cap in Burberry Check pattern.
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A Bastard Box is quite simply any roadside speed camera.
A typical bastard box would be a GATSO speed camera.