I figured this out myself. This will record an entire conversation:
curl -X POST 'https://api.twilio.com/2010-04-01/Accounts/ACc1....4658a/Calls.json' \
-d 'From=%2B44987654321' \
-d 'To=%2B123456789' \
-d 'Record=true' \
-d 'Url=http%3A%2F%2Fourdomain.com%2Ftwiml.php%3Faction%3Dcall_is_taken' \
-u ACc1....4658a:{AuthToken}
The only thing added was the -d 'Record=true'
parameter. I couldn't quite find it in the documentation though.