Regardless of whether you push or pull... What you are doing is inputting pressure in the opposite direction of the turn...
You will turn the bars slightly left, to initiate a right turn,.. and left to initiate a right...
What this does, is put the contact patch of your front tire, from the center of the tire, to one with diminished circumference on in the direction you are turning. Once you are on that part of the tire, and leaned, your bars will shift to turning right in a right turn,... or left in a left...
It's the same reason that a car turning right has slower rotation on right wheels than the left... (if both wheels went the same speed, the car wouldn't turn... it would plow)...
I explained that really badly...

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