Multitasking bar orientation problem on iPad ...

When I double-click the home button, the multitasking bar (tray) shows in correct screen orientation (my game support Landscape Left/Right). But if I switch to another app, and goes back into the game - the multitasking bar locks to Portrait mode. Even if the game still runs in landscape mode... Flipping the orientation does not help.

I have not seen this exact problem. But, check that you are telling Unity to change the orientation properly, with something like the following:

void FixedUpdate () 
{
    // In this example, only the two landscape modes are supported
    if ((Input.deviceOrientation == DeviceOrientation.LandscapeLeft) 
            && (iPhoneSettings.screenOrientation != iPhoneScreenOrientation.LandscapeLeft))
    {   
        iPhoneSettings.screenOrientation = iPhoneScreenOrientation.LandscapeLeft;
    }

    if ((Input.deviceOrientation == DeviceOrientation.LandscapeRight) 
            && (iPhoneSettings.screenOrientation != iPhoneScreenOrientation.LandscapeRight))
    { 
        iPhoneSettings.screenOrientation = iPhoneScreenOrientation.LandscapeRight;
    }
}

You would need code like this in every scene of your game.