Here's my 2 cent worth;
As was stated above, if one can bank the aircraft towards the live engine safely then by default, having banked towards the failed engine, that bank can be reduced back to zero.
While VMCA defines the lowest speed where a test pilot has demonstrated an ability to control a brand new aircraft with the most critical engine failed and the other at maximum power, I as a pilot while being aware of that speed am much more interested in the best rate of climb speed called "blue line speed". It is only at that speed that I can put maximum vertical distance between my aircraft and any obstacles.
However, enough about light twins.
SIDs make no allowance for obstacle clearance. Obstacle clearance is the responsibility of the operator. Thus when departing airports with obstacles, while following the SID on two engines, a B737 will be well capable of clearing all the obstacles. However, if the aircraft suffers an engine failure, the rate of climb will be drastically reduced. If the aircraft is no longer able to clear the obstacles on one engine, the operator will have calculated what is called an "emergency turn".
Example, departing an airport with mountains straight ahead and to the left of the departure track. On two engines, the aircraft can safely climb over the mountains and follow a SID that takes it in that direction (left). However, to allow for the reduced rate of climb, an emergency turn may be specified which will require the aircraft to turn right in the event of an engine failure in order to maintain obstacle clearance at the reduced rate of climb.
Now...if turning into the dead engine can be a problem, how does the CAA allow operators to fly from such airports where regardless of which engine fails, the aircraft will have to turn right?
Regards,
DFC