Many did, many didn't. It often came down to what culture was building the castle, and in what time period.
Most had a dungeon of sort below them, and these could have passages.
A particularly common time for them to have secret passages was in Spain during the inquisition, where there was a "lot of torturing to do" and secret passages would have aided in secrecy of otherwise forbidden/cruel acts.
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Medieval castles.
Medieval castles evolved because the old Motte and Bailey castles where easier to set alight and where not very strong, so they made stone castles to improve defence.
Medieval castles had no steam room.
This big.
Yes, it's true. In medieval castles spiral staircases ascended clockwise.