Is it possible to move a servo without knowing it's current position?
A normal servo is an absolute positioning device. The position of the servo is directly proportional to the length in time of the sync pulse.  This means if you tell the servo to be in a given position, it will seek that position with out regard to the initial position.  It will simply figure out which way to go on it's own and stop when it gets there.

I should add, that in all likely hood, when you manually moved the servo you moved the positioning potentiometer inside the servo.  So it is expected that if the servo is told to go to a particular angle, it will still go to the same angle as it did before you manually moved it.