Either would work, depending on [not according to ] the type of phone it is. You might try Google images to see if you can find a difference.
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HuevosI'd say "put the phone down" or "put the phone back on the rest" or "replace the handset". The only time I hear anyone say "hang up [the phone]" is in a movie.Then you need to come to the US and (apparently) Canada. "Hang up" is very, very common.