You also don't have a right to refuse to answer a police officer generally, such as when they're asking you questions as part of booking you (do you have health conditions they need to know about, for example), and furthermore, during (say, a threshold inquiry or administrative detention) police are legally allowed to treat you refusing to answer them as guilty behavior.
To be clear, in very simplified layman's terms: threshold inquiry = "Hi there, can I ask you a few questions?" and administrative detention = a traffic stop, or "Hi there, come over here, I have some questions to ask you." The primary difference between TI and AD is that in a TI the police ask, and in AD, they order. If you're being ordered to do something, yes, You Are Being Detained, and no, you are not Free To Go. IANAL, etc etc and this shit is annoyingly complex, so please do your own research, folks.
Kafkaesque....try this on for size: thanks to the Berghuis v Thompkins USSC ruling in 2010, it is no longer sufficient to remain silent. You now have to specifically state you wish to remain silent when you are Mirandized.
I wish I were making this up. The court flipped Miranda on its head almost completely. Prior to v Thompkins, the state had to prove, with high standard of evidence, that you had willingly waived your right to remain silent. Now remaining silent isn't enough...and aying anything is taken as an immediate implied waiver.