If all you have is liability you have liability, they won't take your car. But if you have comprehensive, and they're paying you money for damages to the insured vehicle, they will take it. You can buy the car back from them for some miniscule amount.

When my GS got totalled, I moved the car at my expense to a garage from the accident. I got my stereo. I drove my GSX over to it, spent a day swapping interior and body parts, then they came and got it about a week later. I didn't buy it back.

It is your car. You get to tell them if you're going to pull parts from your totalled vehicle, and they can't do anything about it until they've paid you for it. If you haven't seen any money, the car is not theirs yet. Some people let the insurance company tow the car to their storage lot, but once you do, your shit is off-limits because it's private property. Keep the car on your own private property until they show you the money, it gives you more bargaining power and a place to do what you want to it.