Is is possible for Unity 3D smartphone app to support multiple orientations

Is is possible to create a smartphone (iOS and/or Android) app with Unity that supports both landscape and portrait orientations? By support I mean that the app gets notified that the orientation has changed and can adjust layout accordingly.

For example, with the Monkey Island iOS games you play the game in landscape and when you put the game into portrait it displays a menu. Would something like that be possible in Unity?

Yes,

Check Devise Orientation in the documentation. You can continuously check how the device is oriented and change your game accordingly.